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IN  MEMORIAM 
FLORIAN  CAJORl 


Digitized  by  the  Internet  Arciiive 

In  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/fundamentalsofprOOwentrich 


FUNDAMENTALS  OF 
PRACTICAL  MATHEMATICS 


BY 

GEORGE  WENTWORTH 
n 

DAVID  EUGENE  SMITH 

AND 

HERBERT  DEUERY  HARPER 


GINN  AND  COMPANY 

BOSTON    •    NEW  YORK    •    CHICAGO    •    LONDON 
ATLANTA    •    DALLAS    •    COLUMBUS    •    SAN  FRANCISCO 


COPYRIGHT,  1922,  BY  GINN  AND  COMPANY 

ENTERED  AT  STATIONERS'  HALL 

ALL  RIGHTS  RESERVED 


322  3 


GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


\\'^ 


PREFACE 


General  Plan.  The  development  of  a  more  practical  type  of 
education  in  this  country  has  now  reached  such  a  stage  as  to 
demand  a  series  of  textbooks  that  shall  fully  meet  this  need  as 
to  content  and  that  shall  be  prepared  with  the  view  to  their 
usability  in  the  classroom.  In  arranging  to  meet  this  demand 
the  authors  have  made  a  careful  survey  of  both  the  general  and 
the  vocational  schools  in  the  leading  cities  in  the  United  States, 
and  have  come  to  the  conclusion  that  the  most  usable  type  of 
book  should  be  based  upon  the  assumption  that  the  student  has 
had  a  good  course  in  elementary  arithmetic,  including  the  sim- 
ple graph,  but  is  in  need  of  a  brief  review  of  the  fundamental 
operations.    Upon  this  assumption  they  have  proceeded  to  build. 

The  present  book  contains  those  basic  principles  which  the 
student  must  know,  whatever  special  vocation  he  is  to  follow.  He 
may  go  into  machine  work  of  one  kind  or  another,  into  electrical 
work,  into  carpentry,  into  cabinetwork,  into  the  clothing  indus- 
try, or  into  printing,  but  in  any  case  he  will  need  most  of  the 
fundamental  work  which  is  set  forth  in  this  book.  This  work 
consists  of  a  review  of  such  topics  as  the  four  fundamental 
operations  with  integers  and  fractions,  the  practical  use  of  percent- 
age, the  applications  of  proportion,  the  elements  of  mensuration, 
the  use  of  the  formula  and  the  equation,  the  finding  of  areas  by 
plotting  on  squared  paper,  the  finding  of  roots,  and  the  simplest 
elements  of  trigonometry.  The  arrangement  of  the  pages,  with 
the  exercises  facing  the  blueprints,  will  be  found  especially  con- 
venient and  will  contribute  to  the  general  ajjpearance  of  reality 
of  the  work.  As  to  the  exercises  themselves,  they  have  been 
carefully  chosen  from  practical  fields. 

iii 


iv  PREFACE 

Practical  Nature  of  the  Work.  This  review  is  undertaken,  how- 
ever, wi^h  an  entirely  new  set  of  motives  on  the  part  of  the 
student.  Instead  of  mere  mechanical  drill  on  abstract  calculations 
he  at  once  finds  himself  in  the  atmosphere  of  the  shop,  and  he 
meets  with  precisely  the  type  of  problem  that  will  confront  him 
in  his  practical  work.  If  he  has  to  add  fractions  he  will  -find  the 
problem  related  to  a  blueprint  taken  from  the  workroom,  and 
whenever  any  other  operation  is  to  be  performed  the  student  will 
find  that  the  work  always  relates  to  a  real  situation.  Arithmetic 
thus  ceases  to  be  merely  formal  work  with  abstract  numbers, 
algebra  takes  on  an  aspect  of  genuine  utility,  trigonometry  be- 
comes a  tool  to  be  used,  and  mensuration  refers  to  things  that 
the  student  knows  are  worth  measuring. 

Schools  for  which  the  Work  is  Adapted.  As  stated  above,  the 
authors  have  had  in  mind  the  general  high  school  as  well  as  the 
vocational  school.  There  are  many  high  schools  in  which  certain 
classes  will  receive  greater  benefit  from  the  type  of  work  herein 
set  forth  than  from  the  more  abstract  mathematics  commonly 
offered.  The  book  has  therefore  been  prepared  to  meet  the 
needs  of  the  junior  high  school  and  the  four-year  high  school  as 
well  as  the  needs  of  the  technical  and  continuation  schools. 

Technical  Works.  After  completing  the  fundamental  work  laid 
down  in  this  textbook  the  student  will  be  ready  to  take  up  the 
special  preparation  for  his  chosen  vocation.  For  this  preparation 
he  will  need  a  textbook  that  relates  to  the  technical  work  to  be 
undertaken,  such,  for  example,  as  the  "Machine-Shop  Mathe- 
matics" in  this  series.  For  all  such  special  fields  the  present 
textbook  will  be  found  to  give  the  necessary  preparation. 

The  authors  hope  that,  in  preparing  a  work  with  the  same  care 
that  characterizes  all  the  books  of  the  Wentworth-Smith  Series, 
they  have  taken  a  forward  step  in  general  and  vocational  education 
that  will  meet  with  the  same  approval  that  teachers  in  this  country 
have  so  generously  given  to  this  series  in  the  past. 


CONTENTS 

« 

CHAPTER  PAGE 

I.    Fundamental  Operations    ....,>...  1 

II.    Ratio  and  Proportion 55 

III.  Mensuration .  71 

IV.  Trigonometry 131 

V.    The  Slide  Rule 155 

VI.    General  Applications 165 

TABLES  AND  RULES 193 

DEFINITIONS 197 

INDEX .  199 


SYMBOLS 


Symbols.    The  following  mathematical  symbols  and  abbrevi- 
ations are  used  most  frequently  in  the  shop : 


means  3  inches  (3  in.) 

means  2  feet  (2  ft.) 

yard  or  yards 

square  (as  in  sq.  ft.) 

equal,  equals 

angle 

number  (as  in  marking 

•    sizes  of  wires) 

per  (as  in  7  Ib./cu.  ft., 
read  "seven  pounds 
per  cubic  foot") 


Conventional  Signs.    The  following  conventional  signs  are  used 
frequently  in  the  blueprints  upon  which  the  exercises  are  based : 


+ 

plus,  addition 

3" 

— 

minus,  subtraction 

2' 

X 

times,  multiplication ; 

yd. 

by  (as  in2'x  3') 

sq. 

-7- 

divided  by,  division 

= 

V 

square  root 

Z 

6^ 

means  "  5  square," 
or  5  X  5 

# 

: 

ratio  (as  in  2 : 3) 

/ 

2/3 

means  f  or  2  -j-  3 

% 

per  cent,  hundredths 

This  indicates  that  part  of  the  rod  is  omitted  in 
the  drawing. 


¥ 


This  indicates  the  point  from  or  to  which  we  measure. 


Iron    Steel    Brass    Wood 


A  drafting-room   convention  for  cross 
sections  of  the  different  materials  named. 


The  conventional  way  to  show  screw  threads. 


The  conventional  way  to  indicate  an  incandescent  lamp. 


FUNDAMENTALS  OF 
PRACTICAL  MATHEMATICS 


CHAPTER  I 

FUNDAMENTAL  OPERATIONS 

Review  of  the  Fundamentals.  Before  beginning  this  book 
the  student  is  supposed  to  be  able  to  add,  subtract,  multiply, 
and  divide  in  cases  involving  whole  numbers,  decimals,  or 
common  fractions.  Nevertheless,  a  brief  review  of  these 
operations,  undertaken  from  the  strictly  practical  standpoint, 
will  be  found  desirable  unless  the  student  has  recently  been 
doing  a  considerable  amount  of  computing. 

Checks.  Speed  is  desirable,  but  accuracy  is 
essential.    Therefore  cheek  every  operation. 

For  example,  in  this  case  in  addition,  first 
add  from  the  bottom  of  the  column  upwards 
and  then  check  the  result  by  adding  from  the 
top  downwards. 

There  are  several  methods  of  subtracting. 
Use  the  one  that  you  find  best,  but  in  any 
case  always  check  by  adding  the  result  to  the 
smaller  number  and  seeing  that  the  sum  is 
the  larger  number.  In  the  case  of  the  sub- 
traction of  0.638  from  4.07,  as  here  shown,  we 
have  3.432  +  0.638  =  4.07,  and  hence  the  work  is  correct. 

1 


2  FUNDAMENTAL  OPERATIONS 

Exercises.   Addition  and  Subtraction 

1.  In  the  blueprint  of  the  piston  on  page  3  the  lengths 
are  given  to  the  nearest  0.001^'  (thousandth  of  an  inch).  Find 
the  length  A  ;  that  is,  find  the  value  of  1.656"  +  0.063". 

The  expression  1.656''  means  1.656  inches.  Errors  are  less  likely  to 
arise  by  writing  0.063  instead  of  .063,  although  the  two  have  the  same 
meaning.  In  this  book  we  use  the  first  form  except  in  column  addition 
or  subtraction,  although  the  second  form  is  often  used  in  practice. 

2.  In  the  piston  find  each  of  the  lengths  B,  C,  and  Z>. 

3.  Add  the  results  in  Exs.  1  and  2,  thus  finding  the  total 
length  of  the  piston,  and  check  by  adding  the  fourteen 
numbers  that  represent  the  separate  lengths. 

4.  In  the  cone  pulley  find  each  of  the  lengths  A-\-B^ 
A+B  +  C,  and  A  +  B  +  C  +  D. 

5.  In  the  cone  pulley  find  the  length  D  -{-  E  and  the  total 
length  of  the  pulley. 

6.  In  the  cone  pulley  how  much  longer  is  D  than  El 

In  every  case  be  sure  to  check  the  result.  An  incorrect  result  in  a 
simple  operation  of  this  kind  is  never  excusable. 

In  the  lathe  spindle  find  each  of  the  following  lengths : 

7.  A+B,  9.    C  +  D.         11.  E^F.  13.  A^B-\-a 

8.  B+C,         10.  D^-E.         12.  F-\-G.         14.   E-{-F-{-G. 

15.  Find  the  total  length  of  the  lathe  spindle. 

16.  In  the  lathe  spindle  how  much  longer  is  G  than  F? 

In  the  lathe  spindle  find  the  difference  in  length  of  the  parts 
in  each  of  the  following  cases : 

17.  B  and  A,  19.    C  and  D.  21.  E  and  F 

18.  B  and  C.  20.  I>  and  E.  22.  A  and  G. 


ADDITION  AND  SUBTRACTION 


» 


4  FUNDAMENTAL  OPERATIONS 

Multiplication  of  a  Decimal  by  a  Whole  Number.  If  all  the 
numbers  to  be  added  are  equal,  as  in  the  case  of  the  eight 

dimensions  0.375'^  in  the  cross  section  of  the      

piston  on  page  5,  we  can  save  time  by  multi- 
plying. Thus,  we  can  find  the  length  A  by 
finding  the  product,  or  result,  of  8  x  0.375^'. 

In  actual  practice  we  omit   the  abbreviation  for 
inches  ('')  and  arrange  the  numbers  as  shown. 

We  then  have  8x5  =  40,  and  we  write  0  and  add 
4  to  the  next  product.    Then  8  x  7  +  4  =  56  +  4  =  60, 
and  we  write  another  0  and  add  6  to  the  next  product.  Finally,  we  have 
8  X  3  +  6  =  24  +  6  =  30,  and  we  write  30  in  the  product. 

Since  we  multiplied  thousandths  by  a  whole  number,  the  result  is 
thousandths,  and  so  we  have  three  decimal  places.  Since  the  three  deci- 
mal figures  are  zeros,  we  write  3"  as  the  result. 

In  multiplymg  a  decimal  by  a  whole  number^  point  off  from 
the  right  as  many  decimal  places  in  the  product  as  there  are 
in  the  number  multiplied. 

Exercises.   Multiplication  of  Decimals 

1.  Find  the  length  of  four  of  the  parts  of  section  A  of  the 
piston  shown  on  page  5.  Then  multiply  the  result  by  2,  and 
thus  check  the  multiplication  shown  above. 

2.  In  Ex.  1  find  the  length  of  three  of  the  parts. 

3.  In  the  square-head  bolt  the  pitch,  that  is,  the  distance 
between  successive  threads,  is  shown.  Find  the  distance  A; 
the  distance  B;  the  distance  C. 

4.  In  Ex.  3  find  the  distance  ^+^  by  one  multiplication 
and  check  by  adding  the  first  two  results  in  Ex.  3. 

5.  Find  the  width  of  three  lockers  of  the  set  shown  in 
the  blueprint;  of  four  lockers;  of  all  six  lockers. 

6.  Check  the  last  result  found  in  Ex.  5  by  multiplying  the 
first  result  by  2. 


MULTIPLICATION  OF  DECIMALS 


)..)?.?     o.sn    ojr.i  , 


Cf^OSS    SECTION  of  PISTON 


SQUAF\E-HEAD  BOLT 


SET  of  LOCKEf^S 


11.4 
3.14 
456 
114 
342 
35.796 
or  35.8 


6  FUNDAMENTAL  OPERATIONS 

Multiplication  of  a  Decimal  by  a  Decimal.  If  we  have  a  lathe 
faceplate  as  shown  on  page  7,  the  diameter  is  easily  and  accu- 
rately measured  by  the  use  of  calipers  of  some  kind,  but  the 
circumference  cannot  be  so  easily  measured,  since  it  would 
require  the  use  of  a  tape,  which  does  not  give  as  high  a 
degree  of  accuracy  as  a  caliper. 

There  is  a  rule,  however,  that  the  circumference  is  equal 
to  TT  (pi)  times  the  diameter,  where  tt  is  3|, 
or  3.14,  for  ordinary  work,  and  3.1416  for 
cases  requiring  a  higher  degree  of  accuracy. 

Therefore,  to  find  the  circumference  of  the 
faceplate  we  multiply  11.4'^  by  3.14. 

We  first  multiply  as  with  whole  numbers,  the 
product  being  35  796.  Since  we  multiplied  tenths 
by  hundredths,  the  product  is  thousandths;  and  so, 
beginning  at  the  right,  we  point  oif  three  decimal 
places,  the  result  being  35.796. 

Since,  in  this  case,  the  diameter  was  given  only 
to  0.1",  the  circumference  cannot  be  found  accurately  beyond  0.1". 
Therefore  we  write  the  result  as  35.8". 

Beginning  at  the  right.,  point  off  as  many  decimal  places  in 
the  product  as  there  are  in  the  two  numbers  together. 

Exercises.  Multiplication  of  Decimals 

1.  Find  the  dimensions  corresponding  to  those  given  in 
the  collar  pin  on  page  7  for  a  pin  2.5  times  as  large. 

2.  Find  the  dimensions  corresponding  to  those  given  in 
the  cartridge  fuse  for  a  fuse  1.35  times  as  large. 

Midtiply  as  follows : 

3.  2.4  X  3.96.  6.  4.71  x  0.683.  9.  4.9  x  0.087. 

4.  0.57  X  0.873.  7.  52.8  x  37.9.  10.  0.72  x  0.096. 

5.  2.63  X  48.72.  8.  6.82  x  83.5.  11.  3.48  x  5.793. 


MULTIPLICATION  OF  DECIMALS 


8 


FUNDAMENTAL  OPERATIONS 


Division  of  a  Decimal  by  a  Whole  Number.  1.  In  the  bridge 
shown  on  page  9  the  four  struts,  that  is,  the  upright  pieces, 
divide  the  length  into  five  equal  parts.  Find 
the  length  of  each  part. 

Here  we  have  to  find  the  value  of  237.5'  -v-  5. 
Dividing  as  with  whole  numbers,  we  place  the  deci- 
mal point  in  the  result  below  the  decimal  point  in 
the  number  divided.    The  length  of  each  part  is  found  to  be  17.5'. 

2.  Divide  237.5'  by  25. 

In  long  division  we  write  the  result  above  the 
number  divided,  placing  the  decimal  point  in 
the  result  directly  above  the  decimal  point  in  the 
number  divided.   The  result  is  9.5'. 

We  may  check  this  result  by  observing  that 
25  X  9.5'  =  237.5' ;  that  is,  expressed  as  a  general 
rule,  the  number  by  which  we  divide  times  the 
result  is  equal  to  the  number  divided. 


9.5 

25  J)  237.5 

225 

12  5 

12  5 

Exercises.   Division  of  Decimals 

1.  If  the  struts  of  a  steel  bridge  186.5'  long  divide  the 
length  into  five  equal  parts,  how  long  is  each  part? 

2.  If  the  struts  divide  a  bridge  207.2'  long  into  seven 
equal  parts,  how  long  is  each  part? 

3.  In  the  stairway  shown  in  the  blueprint  on  page  9  there 
are  15  steps.    Find  the  rise  R  of  each  step. 

Divide^  finding  each  result  to  the  nearest  hundredth : 

4.  2.9-^4.  7.  0.0728  H- 5.  10.  1^3. 

5.  3.8-^52.  8.  0.0639 -J- 4.  11.  3.1416^4. 

6.  48.1^26.  9.  527.8^649.  12.  100-^3183. 

In  finding  a  result  to  the  nearest  hundredth,  if  the  figure  in  thou- 
sandths' place  is  less  than  5,  it  is  disregarded;  if  the  figure  is  5  or 
greater,  the  figure  in  hundredths*  place  is  increased  by  1. 


DIVISION  OF  DECIMALS 


3/DE  ELEVATION  of  Bf^lDGE 


CnOSS  SECTION  of  3TAIF\  WAY 


10  FUNDAMENTAL  OPERATIONS  | 

Division  of  a  Decimal  by  a  Decimal.   We  often  need  to  find 
the  diameter  of  a  circle  that  shall  have  a  given  circumference. 

Since  circumference  =  tt  x  diameter, 

,                     circumference       ,. 
we  have  =  diameter, 

TT 

or  —  X  circumference  =  diameter. 


For  example,  if  the  circumference  of  the  spur-gear  blan 
shown    on    page   11    is    22.7'',    what    is 
the  diameter  to  the  nearest  0.1"?  ^ 


J 


720 

628 

92 


We  may  divide  22.7"  by  3.1416,  but  it  will  3  11)22  70. 

be  near  enough  for  our  purpose  here  if  we  divide  21  98 

by  3.14. 

To  make  3.14  a  whole  number  we  multiply 
both  numbers  by  100  by  moving  both  decimal 
points  two  places  to  the  right. 

Dividing,  the  result  is  7.2"  to  the  nearest  0.1". 
Since  92,  the  difference  between  720  and  628, 
is  obviously  less  than  h  of  314,  it  will  not  affect  the  tenths'  figure. 

Exercises.    Division  of  Decimals 

1.  Using  3.1416  as  the  value  of  tt,  find  the  value  of  1  ^  tt 
to  the  nearest  ten-thousandth.  Multiply  22.7''  by  the  result 
and  show  that  it  agrees  with  the  7.2"  found  above. 

2.  The  boring-mill  table  shown  on  page  11  has  a  circum- 
ference of  157.38".  By  dividing  by  3.142,  find  the  diameter 
to  the  nearest  0.01".    Check  as  in  Ex.  1. 

3.  The  emery  wheel  has  a  circumference  of  78.54".  Find 
the  diameter  and  the  radius,  using  3.1416  as  the  value  of  tt. 

Divide,  Jtnding  each  result  to  the  nearest  hundredth: 

4.  7^3.142.  6.  5.2-^3.1416.  8.  1-^0.3183. 

5.  6-^3.142.  7.  82.4^3.1416.  9.  5-1-0.3183. 


DIVISION  OF  DECIMALS 


11 


12  FUNDAMENTAL  OPERATIONS 


Exercises.    Review  Drill 

Add  as  follows: 

1.  92.85       2.  62.38 

3.  2.728 

4.  712.3 

5.  9.288 

72.97            93.97 

5.683 

528.2 

.396 

61.82            75.63 

3.929 

887.7 

.486 

32.09            76.06 

6.635 

771.1 

.567 

43.58            29.37 

4.004 

928.6 

8.06 

21.23            82.05 

4.8 

671.8 

9. 

3.89            40.63 

7.303 

466.5 

.079 

Subtract  as  follows  : 

6.  90.24       7.  70.26 

8.  79. 

9.  65.3 

10.  87.03 

30.08            41.9 

32.98 

28.96 

7.6 

Multiply  as  follows: 

11.   3  X  7.04^'.         17. 

1.2  X  3.26. 

23.  6.6  X 

4.92. 

12.   8  X  6.38'.           18. 

56  X  82.2. 

24.  6.2  X 

12.48. 

13.  4  X  5.32".          19. 

9.4  X  3.28. 

25.  0.28  : 

X  2009. 

14.  6  X  3.08'.           20. 

8.1  X  448. 

26.  2.73 

X  816.2. 

15.  9  X  8.754.         21. 

8.5  X  8.23. 

27.  35.5 

X  70.04. 

16.  8  X  0.7382.       22. 

0.34  X  5.814. 

28.  72.39 

>  X  32.016. 

Divide,  finding  each  residt  to  the  7iearest  hundredth : 

29.  428.3-^41.        32.   73.42^0.68.        35.  290.9-4.26. 

30.  720.6-^8.4.       33.  82.37-^7.49.        36.  4.702-32.8. 

31.  293.4^7.3.       34.  64.86^63.8.        37.  53.09-0.772. 

Divide,  finding  each  result  to  the  nearest  thousandth : 

38.  427.1^0.93.        40.  63.85^4.7.        42.  8.030-^23.7. 

39.  5.826 -J- 0.67.        41.  52.27-^6.3.        43.  6.282-42.4. 


REVIEW  EXERCISES  13 

Exercises.   Miscellaneous  Applications 

1.  The  three  sides  of  a  triangle  are  4.75'^  4.9",  and  4.97''. 
Find  the  perimeter ;  that  is,  the  sum  of  the  sides. 

2.  On  four  successive  days  an  automobile  ran  97.6  mi., 
67.8  mi.,  99.7  mi.,  and  82.5  mi.  What  was  the  total  distance? 

3.  From  a  city  lot  that  contained  8635.2  sq.  ft.  the  owner 
sold  3526.8  sq.  ft.  and  2259.3  sq.  ft.    How  much  was  left  ? 

4.  The  perimeter  of  a  triangle  is  19.4",  and  two  of  the 
sides  are  6.48"  and  7.96".    Find  the  third  side. 

5.  If  a  nine-story  city  building  averages  10.75'  to  a  story, 
how  high  is  the  building  ? 

6.'  The  measure  called  the  meter  is  equivalent  to  a  length 
of  39.37".    How  many  inches  are  there  in  6.5  meters  ? 

7.  If  each  side  of  a  square  building  lot  is  66.6',  find  the 
perimeter  of  the  lot. 

8.  A  machinist  made  a  certain  steel  plate  0.75  as  wide  as 
it  was  long.    If  the  length  was  9.2",  what  was  the  width? 

9.  A  certain  rectangle  is  3.45  times  as  long  as  it  is  wide. 
If  the  rectangle  is  6.7'  wide,  what  is  the  length  ? 

10.  How  long  will  it  take  a  railway  train  traveling  0.9  mi. 
a  minute  to  go  29.7  mi.?    to  go  37  mi.? 

In  every  case  where  a  rate  of  speed  is  involved,  the  rate  is  to  be  con- 
sidered as  uniform,  unless  otherwise  stated. 

11.  How  long  will  it  take  an  airplane  traveling  at  the  rate 
of  1.2  mi.  a  minute  to  go  20.4  mi.?    to  go  69  mi.? 

12.  As  already  learned,  l-^7^=  0.3183,  approximately. 
Compare  the  results  of  0.3183  x  13.421"  and  13.421"^ 3.1416, 
finding  each  result  to  the  nearest  0.001".  Which  method  do 
you  find  easier  ?    State  the  reason  for  your  answer. 


14  FUNDAMENTAL  OPERATIONS 

Reduction  of  a  Fraction.  In  practical  mathematics  tlie  com- 
mon fractions  used  most  often  are  halves,  fourths,  eighths, 
sixteenths,  thirty-seconds,  and  sixty-fourths.  By  examining 
a  ruler  divided  into  sixteenths  of  an  inch  the  student  will 
see  that  l"=  %"  =  |'^=  i^e''-    ^^^^'^  ^^®  following: 

1  _2_  4  __8__16  — 32  1  =  A  =  A  =  -1 

2  ~  4  ~  8  ~  16  ~  32  ~"  64  8  16  32  64 
1_2__£_^_16  3_^_12_24 
4~  8  "16  —  32"  64  8  16  32  64 
3_6_12_24_48  6  _  10  _  20  _  40 
4  — 8  — 16  — 32  — 64  8   16   32   64 

From  these  equaUties  we  see  that  the  following  is  true : 

Both  terms  of  a  fraction  may  he  multiplied  hy  the  same 
number  without  changirig  the  value  of  the  fraction. 

Both  terms  of  a  fraction  may  be  divided  by  the  same  number 
without  changing  the  value  of  the  fraction. 

Adding  Fractions.  For  example,  in  the  blueprint  of  the 
shaft  on  page  15  find  the  length  of  A-\-B. 

Since  B  is  given  in  thirty-seconds  of  an  inch,  we 
first  reduce  A  to  thirty-seconds  of  an  inch,  as  shown 
at  the  right.  Since  both  fractions  are  now  expressed 
with  the  same  denominator,  we  can  add  as  shown. 
The  length  of  yl  -f-  i^  is  thus  found  to  be  J|". 


7—14 
16  ~  3  2 
3 
12 

17 
3  2 


Exercises.    Addition  of  Fractions 

In  the  shaft  on  page  15  find  each  of  the  following  lengths : 

1.  B-\-C.  3.  D+E,  5.  A+B-\-C. 

2.  C  +  D.  4.  E-\-K  6.  D-\-E-\-F+G. 

7.  Find  the  total  length  of  the  shaft. 

8.  From  the   blueprint  of  the  tool  post  find  the   length 
oi  A-\-B',  oi  E  +  F;  of  the  tool  post  from  ^  to  i^  inclusive. 

9.  In  the  tool  post  show  that  D +i^=  i|'' =1^/. 


ADDITION  OF  FRACTIONS 


15 


16 


FUNDAMENTAL  OPERATIONS 


Subtracting  Fractions.    1.  In  the  feed  screw  on  page  17  find 
whether  A  or  C  is  the  longer,  and  also  find 
the  difference  in  length  of  the  two  parts. 

We  know  that  -|"  =  i\'',  so  that  A  is  the 
longer.  Since  both  fractions  are  now  expressed 
as  sixteenths  of  an  inch,  we  can  subtract.  AVe 
thus  find  that  A  is   Jg^"  longer  than  C. 

2.  In  the  crankshaft  find  how  much  longer  B  is  than  A. 


If  we  try  to  take  7||"  from 
lllf"  we  find  that  j|"  is  smaller 
than  {f".    So  we  change  11}  f  to 

io-  +  i-+i|- orio-+ir+ir, 

or  10 f  I".    We  can  then  subtract, 
and  we  have  3  j§"  as  the  result. 


HI 


Exercises.    Subtraction  of  Fractions 

1.  In  the  blueprint  of  the  feed  screw  on  page  17  how 
much  longer  is  J^  than  I)  ?  How  much  longer  is  B  than  E  ? 

2.  In  the  crankshaft  how  much  longer  is  B  than  C? 

In  the  crankshaft  find  the  difference  in  length  of  the  parts 
in  each  of  the  following  cases  : 

3.  A  and  Z>.  4.  ^  and  D.  5.  I)  and  C. 

In  the  feed  screw  find  the  difference  in  length  of  the  parts  in 
each  of  the  following  cases  : 

6.  B  and  C,  8.  E  and  C.  10.  E  and  G. 

7.  F  and  G.  9.  E  and  A.  11.  G  and  D. 

In  every  case  express  the  result  in  the  simplest  form.    For  example, 
a  result  like  y%''  should  be  written  ^''. 

12.  In  the  feed  screw  find  the  amount  by  which  the  length 
of  A+B  exceeds  that  of  C-{-D+E  +  E+  G. 


SUBTRACTION  OF  FRACTIONS 


17 


[ 

-— — 

1 

ctzi: 

rJ 

w— 

ZZI^ 

: — L^ 

^ 

^^^ 

^ 

S — 

V- 

— ~] 

-  1 

r         //-^ 


rjo." 

rEED  SCf^EW 


CRANKSHAFT 


18  FUNDAMENTAL  OPEKATIONS 

Multiplication  of  a  Fraction  by  a  Whole  Number.  In  the 
stock-room  bins  shown  on  page  19  it  is  seen  that  the  distances 
between  the  centers  of  the  shelves  are  all  the  same,  namely, 
5^i_''.  To  find  the  sum  of  four  such  distances  we  might 
add,  but  it  is  easier  to  multiply.  Thus,  if  we  wish  to  find 
the  distance  from  the  center  of  the  first  shelf  to  the  center 
of  the  fifth  shelf,  we  have  to  find  the  value  of  4  x  Sg^g"- 

We  multiply  the  fraction  by  4,  and  we  have 
^",  which  is  equal  to  y.  We  then  multiply  the 
whole  number  by  4,  and  we  have  20".  Adding  ^ 
and  20"  we  have  20  J"  as  the  result. 

Instructors  will  find  that  the  detailed  explana- 
tion of  reducing  ^^^  ^^   \  ^^^^  ^^^  ^^^  *^  *^® 
student's  understanding.  With  such  simple  frac- 
tions and  with  the  aid  of  a  ruler  marked  in  fractions  of  an  inch,  the 
student's  intuition  will  lead  him  safely. 

If,  in  some  other  problem,  we  wished  to  find  the  sum  of 
40  distances,  each  b,^-^"  in  length,  we  should  have 

40  X  SgV'  =  2004 1^'  =  200''  +  1'^  +  gV  =  2011''. 

Exercises.   Multiplication  of  Fractions 

1.  In  the  intake  pipe  for  a  6-cylinder  engine,  shown  on 
page  19,  find  the  value  of  2  x  31 1";  find  the  total  distance 
between  centers  for  the  six  cylinders. 

2.  Find  the  height  of  that  portion  of  the  stock-room  bins 
shown  in  the  blueprint,  disregarding  the  thickness  of  the  top 
piece.    Solve  by  one  multiplication. 

3.  In  the  planer  table,  a  part  of  which  is  shown  in  the  blue- 
print, the  distance  between  centers  is  '^-^2!''  Find  by  multi- 
plication the  distance  from  the  first  to  the  ninth  hole ;  from 
the  first  to  the  seventeenth  hole ;  from  the  first  to  the  thirty-third 
hole ;    from  the  first  to  the  forty-ninth  hole. 

The  distance  between  the  centers  of  the  holes  is  always  understood. 


MULTIPLICATION  OE  FRACTIONS 


19 


20 


FUNDAMENTAL  OPERATIONS 


15 

8 

xl5 

8 

_45 

~  8 

2| 
=    5| 
30 
35| 

Multiplication  by  a  Fraction.  1.  In  the  closet  door  shown  on 
page  21  the  length  F  of  the  lower  panel  is  |  the  width  of  the 
door.   Find  P. 

We  have  to  find  |  of  27''.  We  multi- 
ply 27"  by  3  and  divide  the  result  by  4, 
as  shown  at  the  right.  The  length  of 
the  panel  is  thus  found  to  be  20 J''. 

2.  The  height  R  of  the  I-beam  shown  in  the  blueprint  is 
2|  times  the  width.    Find  If. 

We  have  to  find  the  value  of  2^  x  15". 
We  first  find  |  of  15",  as  in  Ex.  1,  and  see 
that  it  is  5f".  Then  2  xl5"=30".  We 
add  these  results  and  find  that  the  height 
of  the  beam  is  35§". 

In  such  a  case  as  that  of  finding  ^  of 
16",  we  first  divide  16"  by  8  and  then  multi- 
ply the  result  by  3. 

3.  In  the  planer  bolt  the  width  W  of  the  head  is  |  the 
diameter  of  the  shank.    Find  W. 

We  have  to  find  j  of  J".  We  first  multiply  the 
two  numerators  together  and  then  multiply  the 
two  denominators  together.  The  width  of  the  head 
is  thus  found  to  be  |^". 

If  the  diameter  of  the  shank  of  the  planer  bolt 
were  ||",  and  the  width  of  the  head  were  f  the 
diameter,  we  should  have  to  find  |  of  |f ".  In  such 
a  case  we  first  indicate  the  multiplication  as  shown, 
cancel,  and  find  the  width  of  the  head  to  be  f ". 

4.  In  the  milling-machine  arbor  the  length  L  is  2|  times 
the  length  of  the  tapered 
part.    Find  L. 

We  have  to  find  the  value 
of  2|  X  3^".  We  change  2 f 
to  -V-,  and  3^"  to  |".  Multi- 
plying as  in  Ex.  3,  we  have  W"",  or  Oy'V".   Hence  the  length  L  is  9^^q' 


2f  X  3-1 

_21x7     147 
8x2       16         i« 

MULTIPLICATION  BY  A  FRACTION  21 


22  FUNDAMENTAL  OPERATIONS 

Exercises.    Multiplication  by  Fractions 

1.  The  width  of  a  door  is  to  be  i  the  height,  and  it  is 
required  that  the  height  be  8^^    Find  the  width. 

2.  The  height  of  an  I-beam  is  23^",  and  the  width  is  one 
third  the  height.    Find  the  width. 

3.  The  circumference  of  a  wheel  is  3^  times  the  diameter. 
If  the  diameter  is  2.1',  what  is  the  circumference  ? 

Find  the  value  of  each  of  the  folloivhuj : 

4.  f  of  1464.              7.  f  of  9.768.  •   10.  2|  x  33.76. 

5.  I  of  327.2.            8.  1  of  7740.6.  11.  3|  x  1.758. 

6.  I  of  77.64.            9.  f  of  7776.6.  12.  7|  x  497.6. 

13.  A  man  has  a  city  lot  tliat  contains  ^^  of  an  acre.  If 
he  sells  |  of  it,  what  part  of  an  acre  does  he  sell  ?  What  part 
of  an  acre  has  he  left? 

14.  If  the  length  of  the  paddle  of  a  water  wheel  in  a  mill 
is  |-  of  the  diameter,  and  the  diameter  is  63'',  what  is  the  length 
of  the  paddle  ? 

15.  If  a  glass  jar  holds  i|  qt.,  what  part  of  a  quart  does 
it  contain  when  it  is  |  full  ?  What  part  of  a  quart  does  it 
contain  when  it  is  |  full  ?  when  it  is  ^  full  ? 

Multiply  as  follows : 

16.  \  X  I-.  18.  I  X  jV  20.  I  X  ^-,.  22.  ^T_  X  f 

17.  A  X  f .  19.   f  X  f .  21.   I  X  If.  23.   ^^  x  ^\, 
24.  Find  the  value  of  |  of  |  of  | ;  the  value  of  |  of  -|  of  -f^. 

Multiply  as  follows  : 

-40.     3    X   g    X  y 5^.  4/.     ^  X  yg    X   g.  A\3.    "5    A    9   X   jg. 

26.  f  X  |1  X  |.  28.  I  X  f  X  ^V  30.  f  X  I  X  12. 


DIVISION  OF  FRACTIONS  23 

Division  of  Fractions.  The  several  cases  involving  the 
division  of  a  fraction  or  the  division  of  a  number  by  a  frac- 
tion, which  the  student  will  meet  in  practical  work,  may  best 
be  understood  by  first  considering  the  following  examples: 

1.  Divide  |  by  3. 

Since  ^  -^  3  is  the  same  as  J  of  J,  we  may  use  multiplication,  as  on 
page  20,  instead  of  division.    We  then  have 

7_i_Q_ly7_      7 

8     •    O  —  -y    X    s   —    24- 

2.  Divide  -If  by  3. 

As  in  Ex.  1,  M  -  3  =  -1  X  }  f  =  y^- 

We  might  have  canceled  the  3  into  the  15,  but  in  such  simple  cases 
as  are  here  given  this  can  best  be  done  mentally. 

3.  Divide  TJ  by  2. 

Since  7^  may  be  written  as  ^^,  we  have 

71-^2  =  ix-V-  =  V-  =  3ii. 

4.  Divide  7i  by  3. 

As  in  Ex.  3,  1\^  3  =  i  x  J/  =  ^  =  H- 

The  method  given  in  Exs.  3  and  4  can  often  be  simplified  by  using 
the  method  shown  in  Exs.  9  and  10  on  page  24. 

5.  Divide  8  by  i. 

On  a  ruler  we  see  that  \"  is  contained  twice  in  1",  and  therefore  \"  is 
contained  8x2  times,  or  16  times,  in  8". 

Hence  8  -f-  i  =  2  x  8  =  1(3. 

6.  Divide  15  by  |. 

As  in  Ex.  5,  we  see  on  a  ruler  that  ^''  is  contained  4  times  in  3". 
That  is,  3"  -4-  f '  =  4. 

Now  15  is  equal  to  5  x  3,  and  therefore 
15  -T-  2  =  5  X  4  =  20. 
We  also  see  that  we  could  get  the  same  result  by  multiplying  15  by  J. 

From  Exs.  5  and  6  we  have  the  following  rule: 

To  divide  hy  a  fraction  multiply  hy  the  fraction  inverted. 


24  FUNDAMENTAL  OPERATIONS 

7.  Using  the  rule  at  the  foot  of  page  23,  divide  |  by  |. 
We  have  |  -.-  §  =  J  x  §  =  ^f^. 

8.  Divide  3|-  by  21 

Expressing  3  J  as  -^g^  and  2^  as  f,  and  canceling  mentally,  we  have 

qi     .91_2v25_5_11 

9.  In  the  line  of  water  pipe  shown  on  page  25  it  is 
found  that  ^  is  J  of  A,    Find  the  length  of  B. 

We  have  to  divide  12  J"  by  2.  Without  expressing 
121''  as  V^  we  can  readily  see  that  12''  -4-  2  =  6",  that 
i"'-4-  2  =  Y,  and  hence  that  B  is  6  J"  long. 

10.  If  it  is  also  found  in  Ex.  9  that  ^  is  |  of  C,  find  the 
length  of  U. 

We  have  to  divide  17 1"  by  4.  If  we 
write  16" +  1  J"  for  17  4",  the  division  is 
easier  as  we  then  have  a  whole  number 
divisible  by  4.  Expressing  IJ"  as  •"/", 
we  see  that  the  length  of  B  is  4^  j". 

Exercises.    Division  of  Fractions 

1.  In  the  line  of  water  pipe  shown  on  page  25  it  is  found 
that  E  is  J  of  F.  Find  the  length  of  JE,  thus  verifying  the 
result  found  in  Ex.  10  above. 

2.  In  the  line  of  conduit  it  is  found  that  JS*  is  i  of  D. 
If  D  is  4'  91"  long,  what  is  the  length  of  ^? 

Express  4'  9|"  as  3'  21 1"  or  as  573". 

3.  Using  the  result  of  Ex.  2,  find  how  many  pieces  of 
conduit  the  length  of  U  can  be  cut  from  a  piece  of  conduit 
8'  long.    How  long  a  piece  is  left  over  ? 

4.  If  ^  in  the  line  of  conduit  is  |  the  length  of  B,  what 
is  the  length  of  ^  ? 

5.  From  Exs.  2  and  4  find  what  part  B  is  of  D, 


DIVISION  OF  FRACTIONS 


25 


26 


FUNDAMENTAL  OPERATIONS 


Fractions  to  Decimals.  Sometimes  the  dimensions  in  a  blue- 
print are  given  in  feet  and  inches,  sometimes  they  include 
a  common  fraction,  and  sometimes  they  include  a  decimal 
fraction.  It  is  necessary  to  be  able  to  express  any  one  of 
these  forms  in  either  of  the  other  two  forms.  We  shall  now 
see  how  to  change  from  a  common  fraction  to  a  decimal  frac- 
tion, or,  as  usually  said,  from  a  fraction  to  a  decimal. 

1.  One  of  the  dimensions  of  the  binding  post  on  page  27 
is  I".    Express  this  as  a  decimal. 

Since  f  means  3  -^  4,  we  simply  divide  3  by  4,  as 
shown.  We  may  annex  as  many  zeros  as  we  wish  after 
a  decimal  point,  for  3  =  3.0  =  3.00,  and  so  on,  just  as 
$a  =  $3.00.  We  see  that  |''  is  equivalent  to  0.75". 

2.  Express  as  a  decimal  the  dimension  ||'^,  which  is  used 
in  the  blueprint  of  the  binding  post. 

We  have  21  ^  64  =  0.328,  with  8  left  over 
from  the  thousandths'  place.  Since  this  8  is 
still  to  be  divided  by  G4,  we  may  write  j?^,  or  ^, 
after  the  thousandths'  place  in  the  result. 

In  practical  measurements  of  this  kind  we 
rarely  need  to  carry  such  a  result  beyond  O.OOI''. 
Since  |  is  less  than  ^,  we  express  f  ^"  as  a  decimal 
to  the  nearest  0.001"  as  0.328". 

If  the  remainder  in  this  case  had  been  ^  or 
greater,  we  should  have  given  the  result  to  the 
nearest  0.001"  as  0.329"  instead  of  0.328". 


0.328^ 

04)21.000 

19  2 

1  80 

1  28 

520 

512 

8 

Exercises.   Fractions  to  Decimals 

1.  Omitting  the  two  dimensions  used  in  the  examples 
above,  express  each  of  the  other  dimensions  of  the  binding 
post  on  page  27  as  a  decimal  to  the  nearest  0.001". 

2.  Express  each  of  the  dimensions  of  the  spark  plug  as  a 
decimal  to  the  nearest  0.001". 


FRACTIONS  TO  DECIMALS  27 


0.437 
64 

1748 
2622 
27.968  (64ths) 

2  8  _  _7_ 
"6  4  "~  16 


28  FUNDAMENTAL  OPEKATIONS 

Decimals  to  Fractions.  Sometimes  a  blueprint  gives  a 
dimension  as  a  decimal  of  an  inch,  and  the  workman  finds  it 
more  convenient  to  use  a  common  fraction  with  a  denominator 
which  is  either  2,  4,  8,  16,  32,  or  64.  If  the  accuracy  of  the 
work  requires  that  dimensions  be  given 
to  Jj'^  he  reduces  the  decimal  to  a  com- 
mon fraction  to  the  nearest  -^^". 

For  example,  in  the  taper  spindle  on 
page  29  the  threaded  part  at  the  top  is 
0.437'^  long.  Express  this  dimension  as 
a  common  fraction  to  the  nearest  J^". 

Since  0.437''  =  ^^^^\  we  can  express  the 
dimension  in  sixty-fourths  of  an  inch  by  mul- 
tiplying both  terms  by  64,  and  we  have  27.968  sixty-fourths,  which  is 
almost  I  J'',  or  /(.".  To  prove  this  we  find  that  7  -^  16  =  0.4375  ;  that  is, 
0.437''  is  too  small  by  0.0005"  to  be  expressed  to  an  exact  ^^". 

Exercises.    Decimals  to  Fractions 

1.  In  the  taper  spindle  on  page  29  express  0.734"  as  a 
common  fraction  to  the  nearest  ^^" ;  to  the  nearest  3^". 

2.  In  the  same  figure  express  each  of  the  other  dimensions 
as  a  common  fraction  or  a  mixed  number  to  the  nearest  -^^", 

3.  In  the  drill  socket  express  each  of  the  dimensions  as  a 
common  fraction  or  a  mixed  number  to  the  nearest  J^'^ 

4.  The  diameters  of  certain  numbered  drills  are  as  follows : 
#15,  0.1800'';  #19,  0.1660'';  #24,  0.1520";  #31,  0.1200". 
Express  each  size  as  a  common  fraction  in  lowest  terms. 

In  the  first  case  reduce  jWoV  t^  lowest  terms  by  first  dividing  by 
100  and  then  by  2.    The  symbol  #  is  commonly  used  for  "  number." 

Reduce  to  common  fractions  or  to  mixed  numbers: 

5.  0.8.         6.  0.35.         7.  1.375.        8.  3.75.        9.  0.625. 


DECIMALS  TO  FRACTIO:NrS 


29 


30  FUNDAMENTAL  OPERATIONS 

Exercises.    Review 

1.  Express  as  a  decimal  each  of  the  dimensions  given  in 
the  blueprint  of  the  dining  chair  shown  on  page  31.  Give 
each  result  to  the  nearest  0.001''. 

2.  Express  each  dimension  of  the  lead -pipe  stop  as  a 
decimal  to  the  nearest  0.001''. 

3.  In  the  figure  of  the  chair  find  the  dimension  lettered  J, 

4.  In  the  same  figure  how  much  greater  is  the  distance 
from  the  top  of  the  front  leg  to  the  top  of  the  back  than 
the  distance  from  the  floor  to  the  top  of  the  front  leg  ? 

5.  In  the  same  figure  find  the  height  II  of  the  back. 

6.  How  long  a  piece  of  wood  is  needed  to  make  four 
pieces,  each  17|"  long  ?  to  make  six  pieces,  each  18|"  long? 
to  make  nine  pieces,  each  3|"  long? 

7.  If  a  child's  chair  is  made  half  as  large  as  the  dining 
chair  shown  in  the  blueprint,  how  long  is  the  piece  corre- 
sponding to  the  one  that  is  17|"  long? 

8.  In  the  pipe  stop  find  the  total  length  oi  A+B  -\-  C -^D. 

9.  In  the  same  figure  find  the  height  E. 

10.  If  the  dimensions  of  a  pipe  stop  were  twice  as  large  as 
those  given  in  the  blueprint,  what  would  be  the  length  of  Z>  ? 

11.  If  the  dimensions  of  a  pipe  stop  were  half  as  large 
as  those  given,  what  would  be  the  length  of  (7? 

12.  Lead  is  HoV  times  as  heavy  as  water,  but  this  fact 
is  usually  stated  by  using  a  decimal.  Express  the  number 
11  ^^Q  with  a  decimal  instead  of  a  common  fraction. 

13.  From  a  piece  of  board  16y  long  a  piece  13|"  long  is 
cut  off.    How  long  is  the  piece  that  is  left  ? 

14.  From  a  steel  rod  261"  long  a  piece  14|"  long  is  cut  off. 
How  long  is  the  piece  that  is  left  ? 


REVIEW  EXERCISES 


31 


32 


FUNDAMENTAL  OPERATIONS 


Exercises.   Review  Drill  Work 

Add  as  follows : 


1.  i  +  J.             3.   7|  +  3J. 

5. 

2i  +  3l. 

7. 

n+n- 

2-i+i             4.21  +  51. 

6. 

5|  +  62. 

8. 

7|  +  9|. 

Subtract  as  follows  : 

9.  A_l.           11.  9|-2i. 

13. 

^-^' 

15. 

83  -  31. 

10.  1  -  |.            12.  9|  -  7|-. 

14. 

Bf-lf 

16. 

^-^' 

Multiple/  as  follows : 

17.   5x  2|.         19.  I|x9. 

21. 

6f  X  4^. 

23. 

2|  X  ^. 

18.  8  X  9|.         20.  6i  X  7i 

22. 

51  X  31. 

24. 

n  X  4|. 

Divide  as  folloivs : 

25.   32 --|.         30.  4|^i. 

35. 

H^h- 

40. 

^i-^i- 

26.   24 -^J^.       31.   2|-^|. 

36. 

H-A- 

41. 

51^42. 

27.   15  ^^2-.       32.   3|-^^. 

37. 

f^2|. 

42. 

1-131. 

28.  f-^J^.^         33.  4f^^V 

38. 

1-31. 

43. 

7f-^l|. 

29.   3^-^l.         34.   2f  ^J^. 

39. 

3f-|. 

44. 

H-H- 

^mt?  ^^g  value  of  each  of  the  following : 

45.  f  of  4  of  1-  of  1  cu.  ft. 

48. 

I  of  1  of  1  of  1  cu.  in. 

46.  i  of  5  X  86  sq.  ft. 

49. 

|ofl5 

x485 

sq.  ft. 

47.  1  of  4  X  164  sq.  in. 

50. 

f  of  13 

xl47 

sq.  in. 

Divide  as  follows : 

51.  21-42.                55.  164. 

-41 

59.  20^-^151. 

52.  48  ^3|.               56.  151 

■^3f. 

60.  301-151-. 

53.  36  ^1|.                57.  171- 

^21. 

61.  27|-^9i. 

54.  24  ^4f                58.  422. 

^10| 

• 

62.  453^151. 

FRACTIONS  33 

Exercises.   Miscellaneous  Applications 

1.  If  a  sheet  of  cardboard  is  ^q"  thick,  how  many  sheets 
pressed  together  will  have  a  thickness  of  1|"  ? 

2.  If  a  sheet  of  veneer  is  ^^q"  thick,  how  many  sheets 
pressed  together  will  have  a  thickness  of  2^"? 

3.  If  a  sheet  of  blotting  paper  is  -^^'^  thick,  how  many 
sheets  are  there  in  a  pile  that  is  12|"  high  ? 

4.  If  a  wagon  wheel  makes  -^^  ^^  ^  revolution  while  the 
wagon  is  going  1',  how  many  feet  will  the  wagon  go  while 
the  wheel  is  making  144  revolutions  ? 

5.  How  many  books,  each  ^"  thick,  will  it  take  to  make 
a  pile  2^''  high  ? 

6.  How  many  strips  of  wood,  each  -^^'^  thick,  will  it  take 
to  make  a  pile  19|l^^'  high  ? 

7.  How  many  sheets  of  bookbinding  board,  each  -^^"  thick, 
will  it  take  to  make  a  pile  9|^'  high  ? 

8.  To  divide  a  sheet  of  paper  7|''  wide  into  four  equal 
columns,  what  width  must  be  spaced  off  for  each  column  ? 

9.  Making  no  allowance  for  doors  and  windows,  how 
much  picture  molding  will  be  needed  to  go  round  a  room 
22|'  long  and  18|^  wide? 

10.  If  you  plan  to  make  a  box  to  go  under  a  shelf  in  a 
space  2iy^  high  and  allow  a  space  of  3i"  between  the  cover 
and  the  shelf,  what  must  be  the  inside  depth  of  the  box  if 
you  use  lumber  ^"  thick  ? 

11.  If  you  are  working  from  a  blueprint  that  gives  the 
dimensions  in  decimals,  and  your  instruments  are  graduated 
in  thirty-seconds  of  an  inch,  what  common  fractions  will 
you  use  in  place  of  the  following:  0.125^',  0.375'^  0.0625^ 
0.1875^  0.625",  0.5625",  0.875",  0.9375"? 


34 


FUNDAMENTAL  OPERATIONS 


Feet  and  Inches.    1.  In  the  border  of  lamps  on  page  35 
find  the  length  of  the  border  between  the  centers 
of  the  first  and  fourth  lamps. 

Since  we  have  to  add  three  dimensions  which  are 
given  in  feet  and  inches,  we  arrange  them  in  columns 
as  shown.  Adding  the  inches  we  have  18",  or  1'  6'',  and 
we  write  6  under  the  inches  column.  Adding  the  feet, 
including  the  1'  already  found,  we  have  7'.  Therefore 
the  required  length  is  7'  6''. 


t 

It 

2 

4 

€> 

9 

1 

5 

7 

6 

2.  The  studs  in  that  part  of  the  partition  shown  in  the 
blueprint  are  equally  spaced, 
distance  between   the 
centers  of   the  studs. 


Find  to  the  nearest  -^^"  the 


We  divide  2r  6"  by  10, 
since  there  are  ten  spaces 
for  the  eleven  studs.  We 
see  that  21'  6"  -f- 10  =  2', 
with  1'  6",  or  18",  left  over, 
which  is  still  to  be  divided 
by  10.  Then  18''  -?- 10  =  1.8". 
the  nearest  y^^",  we  have  \^ 


10)21' 

2' 

of  1' 

^"=2' 

1.8'' 

0.8'^  = 

0.8  of 

16     ■ 

2'  llf - 

or  If" 
.    Arts. 

Expressing  0.8"  as  a  common  fraction  to 
,  and  therefore  the  distance  between  the 


centers  of  the  studs  in  the  partition  is  2'  l|t". 


Exercises.   Feet  and  Inches 

1.  If  the  lamps  on  page  35  were  equally  spaced,  find  to  the 
nearest  \"  the  distance  between  the  centers  of  the  lamps. 

2.  Find  to  the  nearest  -^q"  the  distance  between  the  centers 
of  the  studs  in  a  partition  when  there  are  14  equally  spaced 
studs  in  a  length  of  32';  7  studs  in  a  length  of  18';  18  studs 
in  a  length  of  33';  15  studs  in  a  length  of  28'  7";  12  studs  in 
a  length  of  16'  3";  50  studs  in  a  length  of  75'  10". 

3.  If  the  steel  bar  in  the  blueprint  is  cut  into  five  equal 
parts,  find  to  the  nearest  gL"  the  length  of  each  part. 


FEET  AND  INCHES 


35 


36  FUNDAMENTAL  OPERATIONS 

Exercises.    Review 

1.  The  rivet  holes  in  the  structural  steel  girder,  a  part  of 
which  is  shown  on  page  37,  are  evenly  spaced.  If  the  first 
and  the  last  of  the  holes  shown  in  each  row  are  3'  11^'' 
apart,  find  the  distance  between  successive  holes. 

In  such  cases  distances  are  always  measured  from  center  to  center. 

2.  What  would  be  the  distance  in  Ex.  1  if  there  were  26 
holes  in  the  given  length  ? 

3.  A  machinist  is  laying  out  the  cover  for  the  transmission 
case  shown  in  the  blueprint  for  drilling.  What  distance 
between  centers  does  he  use  for  the  holes  in  the  longitudinal 
rows,  the  holes  being  spaced  equally?  What  distance  does 
he  use  for  the  holes  in  the  transverse  rows  ? 

4.  The  sprinkler  pipe,  a  portion  of  which  is  shown  in  the 
blueprint,  is  2V  10^'  long,  and  there  are  52  equally  spaced 
holes,  the  first  and  last  holes  being  each  ^^^  from  the  ends 
of  the  pipe.    Find  the  distance  between  successive  holes. 

5.  If  the  blueprint  of  the  sprinkler  pipe  in  Ex.  4  had  called 
for  53  holes  with  distances  between  centers  of  4|'',  what 
would  be  the  length  of  the  pipe  ? 

6.  A  certain  blueprint  for  a  steel  girder  calls  for  a  distance 
between  the  centers  of  the  rivet  holes  of  4.375''.  Express  this 
distance  with  a  common  fraction  instead  of  the  decimal. 

7.  If  in  a  girder  the  distance  between  the  first  rivet  hole 
and  the  second  is  3i|''  and  that  between  the  second  and  the 
third  is  4|'',  find  the  distance  from  the  first  hole  to  the  third. 

8.  If  in  a  steel  girder  the  distance  between  the  first  rivet 
hole  and  the  second  is  3|'',  that  between  the  second  and  the 
third  is  4yL-'',  and  that  between  the  third  and  the  fourth  is 
31 1'',  find  the  distance  from  the  first  hole  to  the  third ;  from 
the  second  to  the  fourth;  from  the  first  to  the  fourth. 


REVIEW  EXEECISES 


(o  <>  O  O  <^  O  O  -€>  o  <>  <>  <>  <> 


O  6  O  O  O  -O  -O  <>  O  O  0  O  <> 


5r£"£:z.  GmDEf\ 


TRANSMISSION-CASE  COVEFi 


?i3: 


SPRINKLEF{  PIPE 


38  FUNDAMENTAL  OPERATIONS 

Exercises.   Miscellaneous  Applications 

1.  In  laying  the  floor  of  a  corridor  72'  long,  a  carpenter 
uses  boards  16'  long.  There  are  15  boards  in  the  width  of 
the  corridor.  Allowing  for  the  loss  of  one  board  in  matching, 
how  many  boards  does  the  carpenter  use  ? 

2.  If  the  circumference  of  a  wagon  wheel  is  12',  how 
many  times  will  the  wheel  turn  while  the  wagon  is  going 
14  mi.  ?  while  the  wagon  is  going  7|  mi.  ? 

3.  The  diameter  of  an  iron  rod  is  l|".  Express  this  in 
eighths  of  an  inch.  Is  the  diameter  less  than  i"  ?  If  so,  how 
much  less  ?    Is  it  more  than  |"  ?    If  so,  how  much  more  ? 

4.  A  piece  of  plate  glass  is  -^^"  thick.  Express  this  fraction 
in  lowest  terms.  Express  the  thickness  of  the  glass  in  eighths 
of  an  inch;  in  sixteenths  of  an  inch. 

5.  A  foreman  ordered  some  iron  strips  that  were  ^"  thick 
and  some  that  were  -^2"  thick.  Which  strips  were  the  thicker  ? 
How  much  thicker? 

6.  Which  wire  has  the  greater  diameter,  a  wire  which  is 
l|"  across  or  one  which  is  |"  across  ?    How  much  greater  ? 

7.  A  workman  has  several  drills  which  are  respectively 

¥'^  M"^  If''  tV''  li"'  ^^^^^  ii"  "^  diameter.  Reduce  these 
fractions  to  sixty-fourths  and  arrange  them  in  order  of  size, 
beginning  with  the  smallest. 

8.  If  plaster  |"  thick  is  coated  with  a  finer  plaster  -^~' 
thick,  how  thick  is  the  plaster  then  ? 

9.  A  plate  of  brass  ^^2"  thick  is  laid  on  a  plate  of  iron 
^3g"  thick.    What  is  then  the  total  thickness  of  the  plates? 

10.  An  iron  rod  IJ"  in  diameter  is  covered  with  thin  rolled 
brass  Jg '^  thick.    What  is  then  the  diameter  of  the  rod  ? 

11.  A  piece  of  cardboard  ^j"  thick  is  laid  on  a  book  |" 
thick.    How  thick  are  the  two  together? 


MISCELLANEOUS  APPLICATIONS  39 

12.  How  many  pieces  of  molding  8"  long  can  be  cut  from 
a  strip  5'  long,  and  how  much  molding  will  be  left  over  ? 

13.  Making  no  allowance  for  doors  and  windows,  how 
much  picture  molding  will  be  needed  to  go  round  a  room 
38' 6''  long  and  24'  8''  wide? 

14.  From  a  board  16'  long  a  workman  saws  off  a  piece 
3'  5|"  long  and  another  piece  4' 8|"  long.  How  long  is  the 
piece  of  the  board  that  is  left  ? 

15.  A  plate  of  glass  18|"  by  23|"  was  set  in  a  picture 
frame  that  covered  it  |"  from  each  edge.  What  are  the 
inside  dimensions  of  the  frame? 

16.  A  gas  fitter,  in  running  a  pipe  into  a  room,  has  four 
pieces  of  pipe  respectively  7'  4i ",  8'  2|",  9'  5^ ",  and  8'  6"  long, 
and  finds  that  he  has  4'  9"  more  than  he  needs.  What  is  the 
length  of  the  pipe  required? 

17.  What  is  the  cost  of  16'  4"  of  iron  rod,  weighing  4|  lb. 
to  the  foot,  at  3|(f  a  pound? 

18.  What  is  the  weight  of  a  steel  girder  which  is  18'  10" 
long  and  weighs  46 ^  lb.  to  the  running  foot  ? 

19.  A  pile  of  I -inch  boards  is  5'  3|"  high.  Allowing  5"  as 
the  total  of  the  spaces  left  for  ventilation  between  the  boards 
in  piling,  how  many  boards  are  there  in  the  pile  ? 

20.  The  distance  between  two  railroad  stations  is  given  in 
a  time-table  as  16.32  mi.  Express  this  distance  in  miles  and 
a  common  fraction. 

21.  The  sides  of  a  triangular  flower  bed  are  11'  8",  16'  9", 
and  14'  7"  respectively.    Find  the  perimeter  of  the  bed. 

22.  The  sides  of  a  triangular  plate  of  steel  are  8^^g ",  6|^|", 
and  8^^^"  respectively.   Find  the  perimeter  of  the  plate. 

23.  If  the  perimeter  of  a  triangular  plate  is  23|"  and  if 
each  of  two  sides  is  7^3g",  what  is  the  length  of  the  third  side? 


40  FUNDAMENTAL  OPERATIONS 

Per  Cent.  Another  name  for  "  hundredths  "  is  per  cent.  For 
example,  instead  of  saying  ''  ten  hundredths "  we  may  say 
"  ten  per  cent."    The  two  expressions  mean  the  same. 

There  is  a  special  symbol  for  per  cent,  %.  Thus  we  write 
40%  for  forty  per  cent,  and  it  has  the  same  meaning  as  0.40. 

Because  hundredths  and  per  cents  are  the  same,  any  frac- 
tion with  denominator  100  may  be  written  in  the  form  of 
a  per  cent;   thus: 

4  =  1%       4  =  3%       ^  =  21%       f|5  =  20O% 

The  per  cents  commonly  needed  may  be  easily  written  in 
fractional  form  as  follows: 

5»%  =  ^  =  l  12|%  =  |  37|%=| 

75%=^=!         33|%=i   *       m%=i 

That  part  of  arithmetic  which  treats  of  per  cents  is  called 
percentage. 

First  Problem  in  Percentage.  The  most  common  type  of 
problem  in  percentage  is  finding  some  per  cent  of  a  number. 

For  example,  a  certain  water  tank  holds  85,000  gal.,  and  a 
large  pump  at  the  water  works  pumps  4% 
of   this   amount   at  a  single  stroke.     How 
many  gallons  is  this  ? 


85000 
0.04 
3400.00 
3400 


We  have  to  find  4%  of  85,000  gal.,  and,  as  is 
shown  above,  this  is  the  same  as  0.04  of  85,000  gal. 
Then  4  x  85,000  =  340,000,  and  0.04  x  85,000  is  y  J^ 
as  much,  or  3400.  Therefore  3400  gal.  are  pumped 
at  a  single  stroke. 

In  multiplying  by  any  per  cent,  multiply  as  by  a  whole  num- 
ber and  divide  the  result  by  100  by  moving  the  decimal  point. 
When  it  is  easier,  use  the  fractional  form  of  a  per  cent. 


PERCENTAGE  41 

Exercises.    Finding  Per  Cents 

1.  A  certain  company  manufactures  electric  engines  of 
18,600  H.P.  (horse  power)  and  other  engines  of  66|%  as 
much  power.    Find  the  power  of  the  latter  engines. 

Since  66|%  =  |,  simply  take  §  of  18,600  II. P. 

Most  types  of  engines  or  motors  are  rated  by  the  horse  power  that 
they  develop.  A  horse  power  is  the  force  necessary  to  lift  33,000  lb.  a 
distance  of  1'  in  1  min. 

2.  How  much  does  37|  %  of  a  cubic  foot  of  steel  weigh  if 
1  cu.  ft.  of  steel  weighs  490  lb.  ? 

Use  the  fractional  form  |  instead  of  37^%  or  0.3 7 J. 

3.  If  a  steel  car  when  full  carries  96,000  lb.  of  coal,  how 
many  tons  (2000  lb.)  does  it  carry  when  loaded  to  75%  of 
its  capacity? 

4.  If  a  locomotive  weighing  124  T.  (tons)  can  exert  a  pull 
equal  to  221%  of  its  weight,  how  great  a  pull  can  it  exert? 

Multiply  by  22  J  and  insert  the  decimal  point  two  places  to  the  left. 

5.  If  a  manufacturer  sells  shoes  at  a  profit  of  15%  and  it 
costs  him  $3.45  a  pair  to  make  and  sell  them,  how  much  is 
his  profit  on  1000  pairs  ? 

6.  If  a  shop  manufactures  276  locomotives  and  sells  75% 
of  them  for  |16,125  each  and  the  rest  for  |1 2,825  each,  how 
much  is  received  for  all? 

7.  The  wooden  pattern  from  which  an  iron  casting  is 
made  weighs  6|  %  as  much  as  the  iron.  If  the  casting  weighs 
1500  lb.,  how  much  does  the  pattern  weigh? 

8.  An  iron  tire  expands  ly^g  %  ^^  being  heated  for  shrink- 
ing on  a  wheel.  A  certain  wooden  wheel  needs  a  tire  16'  8" 
in  circumference.    How  much  longer  is  the  tire  when  heated  ? 

Express  16'  8"  as  inches  and  then  multiply. 


42  FUNDAMENTAL  OPERATIONS 

Second  Problem  in  Percentage.   The  second  type  of  problem 
in  percentage  is  to  find  what  per  cent  one  number  is  of  another. 

For  example,  the  purity  of  gold  is  meas- 
ured in  carats,  or  twenty-fourths,  18  carats, 
or  18  carats  fine,  meaning  that  the  article 
is  i|  pure  gold.  What  is  the  per  cent  of 
pure  gold  in  a  14-carat  ring? 


In  a  14-carat  ring  the  pure  gold  is  ^J,  or  j^g,  of 
the  metal.  To  express  yV  ^^  a  decimal  we  divide  7 
by  12,  as  shown  at  the  right,  the  result  being  0.58 
with  a  remainder  of  4  (hundredths).  Dividing  4  by 
12,  we  have  -j*g,  or  J. 

Hence  the  per  cent  of  pure  gold  is  58  J ;  that  is,  the  answer  is  58^%. 

Exercises.    Second  Problem  in  Percentage 

1.  What  is  the  per  cent  of  pure  gold  in  a  watch  case  that 
is  18  carats  fine?    in  a  chain  that  is  10  carats  fine? 

2.  A  carpenter  needs  a  plank  30'  8''  in  length  when  finished. 
To  allow  for  ending  he  orders  it  31'  long  in  the  rough.  The 
waste  is  what  per  cent  of  the  length  in  the  rough  ? 

We  see  that  31'  -  30'  8"  =  4"  and  that  31'  =  372".  We  therefore 
have  to  express  -^^2  ^^  P^^  cent. 

In  all  problems,  unless  otherwise  directed,  if  the  remainder  (hun- 
dredths) does  not  reduce  to  one  of  the  simple  common  fractions,  such  as 
thirds,  fourths,  eighths,  and  so  on,  find  the  decimal  to  the  nearest  0.001 
and  give  the  answer  as  a  per  cent  to  the  nearest  0.1%.  In  practice  it 
might  be  sufficient  to  give  the  answer  to  Ex.  2  to  the  nearest  1%. 

3.  After  a  rough  casting  weighing  282  lb.  is  turned  in  a 
lathe,  it  is  found  to  weigh  271  lb.  The  loss  in  weight  of  the 
casting  is  what  per  cent  of  the  weight  in  the  rough  ?  of  the 
weight  when  finished? 

4.  If  6|  tons  of  iron  are  obtained  from  117 J  tons  of  ore, 
what  per  cent  of  the  ore  is  iron  ? 


f 


PERCENTAGE  43 

5.  An  agent  bought  an  automobile  for  $600  and  sold  it 
at  a  profit  of  $120.  His  gain  was  what  per  cent  of  the  cost? 
of  the  selling  price  ? 

6.  A  cubic  foot  of  water  weighs  62|  lb.  From  a  tank  con- 
taining 800  cu.  ft.  of  water  6250  lb.  of  water  are  drawn  off. 
What  per  cent  of  the  water  is  drawn  off  ? 

7.  If  a  baker  uses  639  lb.  of  flour  in  making  a  certain 
amount  of  bread,  and  adds  to  the  flour  213  lb.  of  liquid, 
the  weight  of  the  liquid  is  what  per  cent  of  the  weight  of  the 
mixture  ?  The  weight  of  the  flour  is  what  per  cent  of  the 
weight  of  the  mixture  ? 

8.  In  preparing  a  solution  for  spraying,  1  oz.  of  Paris  green 
is  added  to  6^  gal.  of  water.  Taking  8.4  lb.  as  the  weight  of 
1  gal.  of  water,  the  weight  of  the  Paris  green  is  what  per  cent 
of  the  weight  of  the  water  ? 

Express  the  result  in  Ex.  8  to  the  nearest  0.01  %. 

9.  A  boiler  that  supplies  steam  for  an  engine  has  a  gage 
attached  to  it  that  shows  how  many  pounds  of  pressure  the 
steam  exerts  against  every  square  inch  of  the  boiler  surface. 
If  the  steam  pressure  increases  from  120  lb.  to  150  lb.  per 
square  inch,  it  is  then  what  per  cent  greater  than  before  ? 

10.  A  plumber  who  has  been  receivmg  $45.60  a  week  for 
a  48-hour  week  has  his  pay  raised  to  $52.80  a  week  for  a 
44-hour  week.    Find  the  per  cent  of  increase  per  hour. 

11.  An  electrical-appliance  dealer  buys  45  box  bells  for 
$16.20  and  sells  them  at  55  (^  apiece.  Find  his  per  cent  of 
gain  on  the  cost;  on  the  selling  price. 

12.  A  rod  that  is  3|'  long  is  what  per  cent  as  long  as  a 
rod  that  has  a  length  of  7f?  The  length  of  the  longer  rod 
is  what  per  cent  of  the  length  of  the  shorter  one?  It  is 
what  per  cent  longer  than  the  shorter  rod  ? 


44 


FUNDAMENTAL  OPERATIONS 


Third  Problem  in  Percentage.  It  often  happens  that  we 
need  to  know  what  number  it  is  that,  when  a  certain  per 
cent  is  taken,  gives  a  certam  other  number. 

For  example,  in  shipping  an  order  of  wet  cells  it  was  found 
that  1050  cells  were  lost  through  breakage,  this  being  15% 
of  the  total  shipment.  How  many  cells 
were  shipped? 

Since  1050=    15%  of  the  total, 

we  see  that  1050  -4-  15  =      1  %  of  the  total, 

and  100  x  1050  -^  15  =  100%  of  the  total. 


7000 

15)105000 
105 


Therefore  we  may  simply  multiply  1050  by  100 
by  annexing  two  zeros,  and  then  divide  by  15,  which  gives  us  7000. 

Hence  there  were  7000  cells  shipped. 

Of  course  we  might  just  as  well  divide  1050  by  0.15,  but  the  above 
method  is  somewhat  clearer  for  purposes  of  explanation. 

The  problem  frequently  appears  in  such  a  form  as  this: 
In  a  shipment  of  wet  cells,  after  losing  15%  in  breakage, 

a  dealer  in  electrical  supplies  could  use  only  5950  cells.    How 

many  cells  were  shipped  to  him? 

Since  he  lost  15%  of  the  cells,  he  lost  j^J^q,  so 
there  were  left  jgg  -  ^V(J>  or  ^%%. 

Since  j^^\  of  the  cells  is  5950, 
^^^  of  them  is  5950  -f-  85, 
and  jgg  of  them  is  100  x  5950  -f-  85. 

Therefore  we  may  simply  multiply  5950  by  100  by  annexing  two 
j:eros,  and  then  divide  by  85,  which  gives  us  7000. 

Hence  7000  cells  were  shipped  to  the  dealer. 

As  in  the  preceding  case,  we  might  divide  5950  by  0.85,  but  the 
method  used  here  avoids  the  use  of  a  decimal. 

Rules  could  be  given  for  this  case,  but  the  student  who 
clearly  understands  the  above  solutions  will  be  able  to  make 
up  his  own  rules  if  necessary. 


PERCENTAGE  45 

Exercises.   Third  Problem  in  Percentage 

1.  Find  the  cost  of  a  D.C.  (direct  current)  generator  which, 
when  sold  at  a  loss  of  8%  on  the  cost,  brought  $575. 

2.  A  cabinetmaker  sold  a  tool  chest  at  a  profit  of  $7, 
which  was  20%  of  the  cost.  Find  the  cost  of  the  tool  chest 
and  also  the  selling  price. 

I        3.  A  clothier  sold  a  suit  of  clothes  at  a  profit  of  20% 
'   on  the  selling  price,  his  gain  being  $8.40.    Find  the  selling 
price  and  the  cost. 

4.  12'  5.6''  is  11%  of  what  length?    5%  of  what  length? 

5.  126  lb.  14  oz.  is  35%  of  what  weight? 

6.  A  hardware  dealer  had  a  bargain  sale  on  a  certain  lot 
of  hammers,  but  failed  to  sell  12^%  of  them.  If  he  had  10 
hammers  left,  how  many  were  there  in  the  lot  ? 

7.  A  man  sells  some  lumber  for  $360,  thereby  gaining 
20%  on  the  cost.  What  per  cent  of  the  cost  is  the  selling 
price  ?    How  much  is  the  cost  ? 

8.  If  a  cabinetmaker  sold  a  desk  so  as  to  gain  20%  on 
the  cost  of  manufacture,  and  received  $84,  how  much  did 
it  cost  to  make  the  desk? 

9.  A  man  saved  $1350.40  last  year,  which  was  32%  of  his 
income.    How  much  was  his  income  ? 

10.  A  contractor  figures  that  of  the  lumber  needed  to  finish 
a  certain  job  he  has  11|^%  on  hand.  If  he  has  on  hand 
23  M  bd.  ft.  (23,000  board  feet),  how  much  does  he  need  to 
finish  the  job  ?    How  much  more  must  he  buy  ? 

11.  A  manufacturer  sold  a  suit  of  clothes  to  a  dealer  at 
a  profit  of  12i%  on  the  cost  of  manufacture.  The  dealer 
sold  the  suit  to  a  customer  for  $48  and  made  a  profit  of  331% 
on  what  it  cost  him.  How  much  did  the  suit  cost  the  dealer  ? 
What  was  the  cost  of  manufacture  ? 


46  FUNDAMENTAL  OPERATIONS 

Exercises.    Miscellaneous  Applications 

1.  An  agent  who  buys  for  an  automobile-supply  company 
receives  from  the  company  a  commission  of  |^%  on  the 
amount  of  his  purchases.  If  he  buys  600  bbl.  of  oil  at  $7.60 
per  barrel,  the  total  freight  amounting  to  |57.50,  find  to  the 
next  higher  25  (f.  the  price  per  barrel  at  which  the  company 
must  sell  the  oil  in  order  to  gain  25%  on  the  total  cost. 

2.  A  hardware  dealer  bought  15  gross  of  screw  drivers  at 
$1.85  a  dozen.  He  sold  three  fourths  of  the  lot  at  a  profit 
of  331%  on  the  cost  and  the  rest  at  a  loss  of  5%  on  the  cost. 
Find  his  per  cent  of  gain  on  the  cost  for  the  whole  transaction. 

3.  A  line  shaft  is  driven  by  a  15  H.P.  motor.  There  is, 
however,  a  loss  of  2i%  due  to  the  slipping  of  the  belt  con- 
necting the  motor  and  the  line  shaft.  Find  the  actual  power 
delivered  to  the  line  shaft. 

4.  A  maintenance  engineer  found  that  in  a  certain  year 
the  per  cent  of  breakage  of  incandescent  lamps  in  a  plant 
was  15%  of  the  number  used.  By  more  careful  placing  of 
the  lights  the  breakage  the  second  year  was  cut  in  half.  If 
in  the  first  year  the  plant  used  4840  lamps,  how  many  did 
it  use  the  second  year  ?  What  was  the  per  cent  of  breakage 
the  second  year? 

5.  A  manufacturer  of  electric  supplies  shipped  300  doz. 
incandescent  bulbs  to  a  jobber.  Owing  to  an  accident  in 
transportation,  15%  of  the  bulbs  were  broken.  How  many 
bulbs  were  broken  ?    What  per  cent  were  not  broken  ? 

6.  The  profits  on  a  business  this  year  are  $12,688,  and  are 
22%  more  than  last  year.    What  were  the  profits  last  year? 

7.  Water  in  freezing  expands  9%  of  its  volume.  How 
many  cubic  feet  of  water  are  needed  to  make  1199  cu.  ft.  of 
ice  ?    How  many  gallons  (231  cu.  in.)  of  water  are  needed  ? 


DISCOUNT 


47 


Discount.  Merchants  who  sell  at  wholesale,  that  is,  in  large 
quantities  to  dealers,  often  sell  at  a  certain  per  cent  off  the 
list  price^  as  the  price  stated  in  their  catalogues  is  called. 

A  reduction  from  a  price  or  amount  is  called  a  discount 

The  per  cent  of  discount,  or  the  common  fraction  to  whicli 
this  per  cent  is  equivalent,  is  called  the  rate  of  discount. 

The  amount  of  a  price  after  the  discount  has  been  taken 
off  is  called  the  net  price  or  selling  price,  and,  similarly,  the 
amount  of  a  bill  after  the  discount  has  been  taken  off  is  called 
the  net  amount  of  the  bill. 

For  example,  find  the  discount  and  the  net 
price  when  the  list  price  of  an  adding  ma- 
chine is  $245  and  a  discount  of  12%  is  allowed 
the  purchaser. 

We  first  take  12%  of  $245  and  find  that  the  dis- 
count is  $29.40. 

We  then  subtract  the  discount  from  the  list  price 
and  find  that  the  net  price  is  $215.60. 

In  cases  where  the  rate  of  discount  can  be  expressed  easily 
as  a  simple  common  fraction,  it  is  better  to  use  that  fraction. 

For  example,  find  the  discount  and  the 
net  amount  of  a  bill  of  goods  for  $1248, 
the  rate  of  discount  bemg  16|%. 

We  first  express  the  rate  of  discount  as  a  frac- 
tion, 16 §%  being  equivalent  to  the  fraction  ^. 

We  then  find  the  discount  by  taking  f  of  $1248 
and  obtain  $208  as  the  result. 

We  then  subtract  this  discount  from  the  amount 
of  the  bill  and  find  that  the  net  amount  is  $1040. 

Although  there  is  no  fixed  custom  in 
business,  in  solving  problems  in  this  book  the  student  should 
discard  any  fraction  less  than  i(f  in  a  discount  and  call  i^ 
or  more  a  full  cent. 


48 


FUNDAMENTAL  OPERATIONS 


Discounted  Bill.    A  bill  sent  by  a  wholesale  dealer  usually 
shows  the  discount  allowed.    A  sample  receipted  bill  follows : 


Chicago,  111.,  Dec.   17,  1924 

mv.  B.  S.  Cole 

Oklahoma  City,  Okla. 

^"S*'"^     HILL,  SMITH  &  CO. 

MANUFACTURING  JEWELERS 

8378  Burlington  Ave, 
Terms:  2/10,    net  30 

Dec. 

17 

3  doz.  silver  forks      @  S32 
1/6  doz.  salad  forks    @  ^33 

Less  20% 

RECEIVED  PAYMENT 

DEC.  22,  1924 

HILL,  SMITH  &  CO. 

Per               <^,.^M.. 

96 

5 

50 

101 
20 

.50 
30 

81 
/ 

79 

20 

6S 

A  bill  sent  by  a  manufacturer  or  a  jobber  is  often  called 
an  invoice.    The  above  bill  might  be  called  an  invoice. 

The  expression  "  Terms:  2/10,  net  30  "  in  the  above  bill 
means  that  Mr.  Cole  may  take  off  an  additional  2%  discount 
if  he  pays  the  bill  withhi  ten  days  or  that  he  may  have  thirty 
days  in  which  to  settle  the  account  for  the  net  amount, 
$81.20.  When  Mr.  Cole  received  the  bill  it  showed  the  net 
amount  after  the  20%  discount,  or  trade  discount  as  it  is 
often  called,  had  been  taken  off.  When  Mr.  Cole  paid  the 
bill  he  deducted  the  2%,  as  shown  in  the  script  type. 


DISCOUNT  49 

Exercises.    Bills  and  Discounts 

Make  out  hills  for  each  of  the  following,  take  off  the  discounts 
given,  and  write  the  receipt  on  each  hill  as  on  page  48 : 

1.  2  carloads  coal,  23,800  lb.,  25,200  lb.,  @  |9.80  per  ton; 
3  carloads  coal,  24,800  lb.,  23,700  lb.,  22,900  lb.,  @  |9.95  per 
ton.    Terms:  3/10,  net  90. 

In  all  these  problems  insert  the  dates  and  the  names  of  dealer  and 
purchaser,  and  assume  that  the  bill  is  paid  within  the  time  allowed  for 
the  cash  discount,  as  the  3  %  in  10  da.  is  called. 

In  all  problems  consider  the  ton  as  2000  lb.  unless  the  long  ton  of 
2240  lb.  is  specifically  mentioned. 

2.  7  rocking  chairs  @  $8.30;  9  kitchen  tables  @  |2.75; 
3  bedroom  sets  @  $54.30 ;  18  extension  dining  tables 
@   $16.50;  9  sideboards  @   $24.60.   Terms:   2/10,  net  30. 

3.  1 T.  fence  wire  @  23  ^  per  pound.  Trade  discount :  35%- 
Terms:  2/10,  net  60. 

4.  1500  lb.  of  3d.  nails  @  $3.67  per  100  lb.  Trade  dis- 
count: 15%.    Terms:  3/10,  net  30. 

5.  45  bu.  of  hair  for  plastering,  8  lb.  to  the  bushel,  @  9(f 
per  pound.    Trade  discount:  12%.    Terms:  3/10,  net  60. 

6.  2  automobile  cylinders  complete  with  head  cover, 
valves,  and  plugs  @  $69.70.    Trade  discount:  8%. 

7.  20 J  doz.  ^^g-inch  cylinder-head  cover-cap  screws  @  900 
per  dozen.    Trade  discount:  12%.    Terms:  2/10,  net  60. 

8.  5892'  #18  annunciator  wire,  150'  to  the  pound,  @  53^ 
a  pound.    Trade  discount:  7|^%. 

9.  121b.  black  rubber  tape,  -J-pound  rolls,  @  15(f  a  roll. 
Trade  discount:  J.    Terms:  2/10,  net  30. 

10.  11,646'  #14  galvanized  telephone  wire,  96  lb.  to  the 
mile,  @  7|0  a  pound.    Trade  discount:  ^. 


60  FUNDAMENTAL  OPERATIONS 

Several  Discounts.  In  some  lines  of  wholesale  business  two 
or  more  discounts  are  occasionally  allowed  on  the  same  bill. 

For  example,  a  dealer  may  buy  |400  worth  of  hardware 
with  discounts  of  20%  and  10%,  often  written  "20%,  10%," 
or  simply  "20,  10."  This  means  that  20%  is  first  deducted 
from  the  amount  of  the  bill  and  then  10%  is  deducted  from 
the  remainder.    What  is  the  net,  amount  of  the  bill  ? 

The  amount  of  the  bill  is  $400. 

The  amount  less  20%  is  |320. 

The  $320  less  10%  is  |288,  the  net  amount. 

This  explains  the  general  nature  of  the  problem.  Practi- 
cally we  should  take  short  cuts.  If  we  take  away  20%  of  an 
amount  we  have  80%  left,  and  if  we  take  away  10%  of  this 
result  we  have  90%  of  it  left.  The  problem  becomes  a  case 
of  finding  90%  of  80%  of  |400,  and  we  have 

0.90  X  0.80  X  1400  =  0.72  x  |400  =  |288. 

Proceeding  in  the  same  way  we  can  solve  a  problem  of  this 
type  when  it  is  easier  to  use  the  fractional  form  of  the  per 
cents.  If  the  above  discounts  were  12i%,  16|%,  we  should 
have  to  find  |  of  |  of  $400,  and  we  should  have 

25 

^Al2SJ^=M5  =  $291,661. 
^  X  ^  3 

3 

Therefore  the  net  amount  would  be  $291.67. 

Follow  the  rule  given  at  the  bottom  of  page  47  in  connection  with 
fractions  of  a  cent  in  any  discount. 

The  bill  for  the  above  goods  sent  by  the  wholesaler  would  follow  the 
form  of  the  discounted  bill  shown  on  page  48.  If  any  cash  discount  is 
allowed,  it  is  deducted,  when  the  account  is  settled,  from  the  net  amount 
shown  on  the  wholesaler's  bill. 


SEVERAL  DISCOUNTS  51 

Exercises.   Several  Discounts 

1.  What  is  the  difference  between  a  discount  of  50%  and 
the  two  discounts  of  25%,  25%  on  $1000? 

2.  Is  there  any  difference  between  a  discount  of  5%,  4% 
and  one  of  4%,  5%  on  $900?   on  $600? 

3.  A  manufacturer  hsts  a  desk  at  $52  less  25%,  and  a 
rival  manufacturer  offers  a  similar  desk  for  $57  less  i.  Which 
is  the  lower  net  price  ?  How  much  lower  ?  If  the  first  dealer 
increases  his  discount  to  25,  3,  which  will  then  be  the  lower 
net  price  ?    How  much  lower  ? 

4.  A  wholesale  dealer  allows  a  trade  discount  of  20,  5,  and 
a  10-day  cash  discount  of  2%  from  his  list  prices.  Find  the 
amount  paid  on  Jan.  18  for  the  following  items  bought  on 
Jan.  10  :  9  dining  tables  @  $34.50  ;  68  dining  chairs  @  $7.85  ; 
7  buffets  @  $47.25 ;  5  serving  tables  @  $29.75. 

5.  Make  out  an  invoice  for  the  following  goods  bought 
Sept.  20  and  paid  for  Oct.  5,  terms  2/20,  N  90:  5  #264 
plows  @  $42.50  less  20%  ;  3  #178  self -dumping  hayrakes  @ 
$18.60  less  15%  ;  9  #325  hay  stackers  @  $46.50  less  15%. 

The  symbol  2/20,  N  90  means  the  same  as  2/20,  net  90,  the  explana- 
tion of  which  was  given  on  page  48. 

6.  Find  the  amount  paid  for  2469^  of  100-conductor  in- 
terior cable,  1|-'  to  the  pound,  @  54^  a  pound,  net. 

Following  the  model  on  page  48,  make  out  a  receipted  bill  for 
each  of  the  following  shipments  : 

7.  36doz.  files  @  $8.30.    Discount:  30,  20. 

8.  9  doz.  pairs  hinges  @  $6.75 ;  48  doz.  table  knives  @ 
$12.60;  36  doz.  table  forks  @  $8.40.    Discount:  20,  10. 

9.  36  doz.  locks  @  $6.30;  19  doz.  mortise  locks  @  $6.75; 
28  doz.  pairs  hinges  @  $8.85.   Discount:  25,  8,  4. 


52 


FUNDAMENTAL  OPERATIONS 
Exercises.   Pay  Rolls 


1.  Fill  out  each  space  that  is  marked  witli  an  asterisk  (*) 
in  the  following  form,  which  is  part  of  the  pay  roll  of  a  small 
manufacturing  concern  for  the  week  ending  July  26,  1924: : 


PAY  ROLL  OF  J.  R.  MOLLER  &  CO. 

For  the  week  ending  July  26,  7924 

No. 

Name 

M. 

T. 

W. 

T. 

F. 

S. 

Total 
Time 

Wages 
per  Hour 

Total 
Wages 

1 

T 

J.  P.  Drew 

8 

8 

8 

6 

8 

4 

42 

80)^ 

S33 

60 

R.  L.  Bond 

8 

-rl 

8 

8 

8 

4 

* 

85^ 

* 

* 

3 

P.  F.  Cram 

7 

7i 

8 

■^i 

8 

4 

* 

80^ 

* 

* 

4 
5 

B.  J.  Mead 

8 

7 

6J 

8 

^i 

3 

* 

75^ 

* 

* 

R.  K.  King 

8 

8 

7 

n 

8 

3i 

* 

72^ 

* 

* 

6 

M.  L.  Drake 

^ 

6 

8 

0 

7 

3i 

* 

67  J/' 

* 

* 

* 

* 

* 

* 

* 

* 

* 

* 

* 

Make  out  pay  rolls  (inserting  names')  when  the  men's  num- 
bers,  the  hours  per  day,  and  the  wages  per  hour  are  as  follows : 

2.  No.  1:  71  71,  7},  8,  8,  31,  90  (f;  No.  2:  8,  8,  8,  8,  8, 
4,  85(f ;  No.  ?>\  8,"'8,  7,  7,  8,  4,  72(^;  No.  4:  8,  71,  6|,  8,  8, 
4,  68(^;  No.  5:  8,  8,  8,  0,  61,  4,  65(f. 

3.  No.  1:  7,  8,  8,  8,  71,  4,  92i(^;  No.  2:  8,. 7,  8,  7,  6,  4, 
80(f;  No.  3:  8,  71    8,  8,  8,  4,  1^^\  No.  4:  8,  8,  8,  8,  71, 

4,  750;  No.  5:  8,^7,  8,  8,  8,  4,  12^. 

4.  No.  1:  8,  7,  6,  8,  8,  4,  9210;  No.  2:  8,  7,  6,  6|,  71 
4,  840;  No.  3:  8,  6,  6,  8,  8,  4,  820;  No.  4:  7,  8,  7|,  61, 
8,  4,  770;  No.  5:  8,  6,  7,  71   6i,  4,  680. 

5.  No.  1:  7,  7,  8,  8,  8,  4,  960;  No.  2:  8,  8,  7|,  71  7|,  4, 
950;  No.  3:  8,  8,  8,  7|,  7|,  4,  880;  No.  4:  71,  7{,  71,  8, 
7f,  3|,  820;  No.  5:  8,  8,  8,  8,  8,  4,  7210. 


PAY  EOLLS 


63 


6.  Fill  each  space  marked  with  an  asterisk  in  the  following 
pay  roll,  allowing  double  pay  for  overtime  as  explained  below: 


PAY  ROLL  OF  R.  E.  THURSTON  &  CO. 

For  the  week  ending  Jan.  19,  7924 

No. 

Name 

M. 

T. 

w. 

T. 

F. 

S. 

Total 
Time 

Wages 
per  Hour 

Total 
Wages 

1 

R.  S.  Jones 

v 

V 

v/ 

^t/ 

i 

s/ 

54 

90  0 

* 

* 

2 

M.  L.  Downs 

V 

V 

V 

i 

6i 

'1/ 

* 

85^ 

* 

* 

3 

J.  M.  Reed 

>/ 

V 

?/ 

1 

^ 

* 

82\f 

* 

* 

* 

* 

* 

* 

* 

* 

* 

* 

* 

Before  assigning  Ex.  6  the  instructor  should  explain  that  from  one 
and  a  half  to  two  times  the  regular  hourly  wage  is  usually  paid  for 
overtime,  and  that  the  check  (  v/)  in  the  above  pay  roll  means  full  time 
for  the  day.  In  this  pay  roll  the  full  time  is  8  hr.  except  on  Saturday, 
when  it  is  4  hr.  The  symbol  1/  means  8  hr.  +  2  hr.  overtime.  A  dash 
( — )  indicates  absence.  Part  time,  like  Gi^hr.,  is  indicated  as  above  on 
Friday  for  Downs.  Since  the  allowance  for  overtime  is  double  that  for 
regular  work,  Jones's  time  is  8  +  8  +  8  +  8  +  8  +  4  (regular  time)  and 
4+3  +  3  (overtime),  or  54  hr.  in  all. 

Make  out  'pay  rolls  (inserting  names')  when  the  merCs  numbers, 
the  hours  per  day,  and  the  wages  per  hour  are  as  follows,  a  full 
day  being  8  hr.  except  on  Saturday,  when  it  is  4  hr.,  and  double 
pay  being  given  for  overtime  : 

7.  No.  1:  8,  9,  8,  9,  8,  5,  97i(^;  No.  2:  8|,  9,  9-},  8,  8, 
4,  95(f ;  No.  3:  8,  8,  8,  10,  8,  6,  92^ ;  No.  4:  8,  9,  9,  9,  7,  4, 
90  (f;  No.  5:  81    81,  9,  8,  8,  5,  80^. 

8.  No.  1:  8,  10,  8,  10,  8,  6,  90(f ;  No.  2:  91,  8,  6,  9,  8,  4, 
88(f;  No.  3:  10,  10,  10,  10,  8,  5,  821(^;  No.  4 :  8,  0,  8,  8, 
10,  9,  80(ife;  No.  5:  8,  8,  8,  9,  81,  6|,  75^. 


54  FUNDAMENTAL  OPERATIONS 

Exercises.    Review 

1.  A  casting  is  14'  3"  long,  and  the  metal  expands  -^^"  to 
the  foot  when  heated  to  a  red  heat.  Compute  the  length  of 
this  casting  when  red  hot. 

2.  In  casting  brass  hinges  an  allowance  of  -J^  in  length  has 
to  be  made  for  shrinkage  when  the  brass  cools.  Find  to  the 
nearest  0.001^'  the  length  of  the  mold  for  a  hinge  that  is  to 
be  3y  long;  for  a  hinge  that  is  to  be  5|''  long. 

3.  Find  the  weight  of  a  piece  of  round  steel  shafting  18|' 
long,  weighing  24.05  lb.  to  the  running  foot. 

4.  Find  the  cost  of  780'  of  #  000  stranded  rubber-covered 
wire  @  1375  per  1000',  discounts  of  12%,  5%  being  allowed. 

5.  Find  the  cost  of  967'  of  |-inch  rigid  conduit  @  $16.15 
per  100',  discounts  of  8%,  3%  being  allowed. 

6.  A  manufacturer  determines  his  selling  price  by  adding 
approximately  22^%  to  the  manufacturing  cost.  If  it  costs 
$72.50  to  manufacture  a  certain  machine,  iind  to  the  next 
higher  dollar  the  price  at  which  he  should  sell  it. 

7.  A  furniture  dealer  buys  some  tables  at  a  prime  cost  of 
$16.75  each.  His  buying  and  selling  expenses  and  overhead 
total  24%  of  the  prime  cost.  If  he  sells  the  tables  at  $25  each, 
what  per  cent  of  profit  does  he  make  on  the  prime  cost? 
on  the  total  cost? 

The  first  cost,  or  wholesale  net  price,  of  an  article  is  called  the  prime 
cost.  Such  expenses  as  heat,  light,  insurance,  and  so  on  make  up  the 
overhead  charges,  which  are  often  called  simply  overhead. 

8.  A  dealer  bought  some  glassware  for  $578.50,  but  through 
his  fault  some  of  the  glassware  was  broken.  He  sold  the  rest 
for  $600,  and  estimated  his  cost  of  doing  business  at  22% 
of  the  selling  price.  Did  he  gain  or  lose  on  the  transaction  ? 
What  per  cent  of  the  total  cost  did  he  gain  or  lose  ? 


CHAPTER  II 

RATIO  AND  PROPORTION 

Ratio.  In  practical  mathematics  we  make  much  use  of  the 
idea  of  ratio,  as  of  the  ratio  of  one  length  to  another  length. 
The  relation  of  one  number  to  another  of  the  same  kind,  as 
expressed  by  the  division  of  the  first  number  by  the  second, 
is  called  the  ratio  of  the  first  number  to  the  second. 

Thus 'the  ratio  of  $3  to  $6  is  J,  or,  in  its  simplest  form,  I  'i  the  ratio 
of  1  yd.  to  1  ft.  is  the  same  as  the  ratio  of  3  ft.  to  1  ft.,  or  3  ;  the  ratio 
of  5  to  2  is  f ,  or  2  J ;  and  the  ratio  of  any  number  to  itself  is  1. 

The  ratio  of  2  to  3  may  be  written  in  the  fraction  forms 

2  or  2/3,  or  in  the  form  2 ;  3. 

12'    12 
The  ratio  of  12'  to  4',  for  example,  may  be  written  — ,  — ,  12/4, 

4'      4 
12  : 4,  or  simply  3.    The  word  "ratio  "  is  used  for  each  of  these  forms. 
The  expression  12  : 4  is  read  "the  ratio  of  12  to  4  "  or  "as  12  is  to  4," 
12  and  4  being  called  the  terms  of  the  ratio. 

Illustrative  Problem.  If  the  driving  wheel  of  a  locomotive 
has  a  diameter  of  6'  9''  and  a  circumference  of  21'  2.47'',  find 
the  ratio  of  the  diameter  to  the  circumference. 

To  find  the  ratio  we  have  to  divide  the  diameter  by  the  circum- 
ference, and  in  this  case  we  have 

^:i^=-ii:i-= 0.3183. 

21' 2.47"      254.47" 
Hence  the  ratio  of  the  diameter  to  the  circumference  is  0.3183. 

The  above  value  is  the  same  as  that  obtained  on  page  10 
by  dividing  1  by  3.1416  ;  that  is,  in  any  circle  the  ratio  of 
the  diameter  to  the  circumference  is  1  -^  tt. 

55 


56 


RATIO  AND  PROPORTION 


Ratios  and  Scales.  One  of  the  simplest  uses  of  ratio  is 
found  in  the  blueprints  commonly  seen  in  shops. 

For  example,  if  a  room  is  30^  long  and  20^  wide,  and  we 
make  a  floor  plan  3'^  long  and  2'^  wide,  we  draw  the  plan  to 
scale,  1^'  representing  10'.  We  indicate  this  by  writing 
"Scale,  l''=10'."  We  may  also  write  this  "Scale,  l''  =  120'V' 
or  "  Scale  yJo"'"  which  simply  means  that  the  ratio  of  the 
length  of  a  line  in  the  drawing  to  the  length  of  the  line  on 
the  floor  which  it  represents  is  1 :  120. 

The  following  shows  a  line  AB  drawn  to  different  scales : 

A  B 


J    Scale  J 

J    Scale  ^ 
1 


The  figures  shown  below  illustrate  the  drawing  of  a  rec- 
tangle to  scale.  In  this  case  the  lower  rectangle  is  a  drawing 
of  the  upper  rectangle  to 
the  scale  ^,  or  1  to  2,  or 
1''  to  2". 

Notice  that  the  area  of  the 
lower  rectangle  is  only  ^  that 
of  the  upper  one;  that  is,  the 
ratio  of  the  areas  is  1 : 4.  When 
we  draw  to  the  scale  \  we  mean 
that  the  length  of  every  line  in  the  drawing  is  \ 
the  length  of  the  corresponding  line  in  the  original. 
Whatever  the  shape  of  the  figure,  the  area  of  the 
drawing  is  \  the  area  of  the  original  figure. 

Maps  are  figures  drawn  to  scale.  The  scale  is  usually 
stated  on  the  map,  as  you  will  see  in  any  geography.  The 
scale  used  in  drawing  a  map  is  often  expressed  by  means 
of  a  line  divided  to  represent  miles,  and  sometimes  by  such 
a  statement  as  "  1"  =  100  mi." 


DRAWING  TO  SCALE 


57 


Exercises.   Drawing  to  Scale 

1.  A  room  is  30'  long  and  25'  wide.    Draw  a  plan  of  the 
floor,  using  the  scale  j^-^. 

2.  Draw  a  plan  of  a  floor  24'  long  and  16'  wide,  using 
the  scale  of  ^"  to  1';  using  the  scale  of  1"  to  8'. 


TORCH 


3.  The  drawing  above  is  the  floor  plan   for  a  concrete 
bungalow.    Find  the  scale  used  in  drawing  the  plan. 

4.  Find  the  dimensions  of  the  living  room,  dining  room,  and 
smaller  bedroom,  including  the  wardrobe  but  not  the  closet. 


58  RATIO  AND  PROPORTION 

Exercises.   Miscellaneous  Applications 

1.  The  specific  gravity  of  copper  is  8.9  and  the  weight  of 
1  cu.  ft.  of  water  is  621  lb.  Find  the  weight  of  1  cu.  ft.  of 
copper ;  of  7.53  cu.  ft.  of  coppeV. 

The  specific  gravity  of  a  metal  is  the  ratio  of  the  weight  of  a  piece  of 
the  metal  to  the  weight  of  an  equal  volume  of  water. 

2.  The  specific  gravity  of  tin  is  7.29.  Find  to  the  nearest 
0.1  oz.  the  weight  of  1  cil.  in.  of  tin. 

First  find  the  weight  of  1  cu.  ft.  of  tin,  using  the  weight  of  1  cu.  ft. 
of  water  as  given  in  Ex.  1.    Then  1  cu.  in.  =  j^Vg  cu.  ft. 

3.  The  weight  of  1  cu.  in.  of  water  is  0.58  oz.,  and  the 
specific  gravity  of  steel  is  7.83.  Find  the  weight  of  1  cu.  in. 
of  steel ;  of  1  cu.  ft.  of  steel. 

4.  Using  the  specific  gravity  of  copper  given  in  Ex.  1,  find 
the  weight  of  a  bar  of  copper  8''  long,  2"  wide,  and  1"  thick. 

5.  In  Babbitt  metal,  also  known  as  white  metal,  there  are, 
by  weight,  4  parts  of  copper,  8  parts  of  antimony,  and 
96  parts  of  tin ;  that  is,  the  ratio  of  the  copper  to  the  total 
is  4 :  108.  In  54  lb.  of  Babbitt  metal  what  is  the  weight  of 
the  copper  ?  of  the  antimony  ?  of  the  tin  ? 

6.  From  the  data  of  Ex.  5  find  the  weight  of  the  copper 
in  500  lb.  of  Babbitt  metal. 

7.  From  the  data  of  Ex.  5  find  the  weight  of  each  of  the 
three  component  metals  in  800  lb.  of  Babbitt  metal;  in 
900  lb.  of  Babbitt  metal ;  in  1  T.  of  Babbitt  metal. 

8.  A  certain  kind  of  gunpowder  is  made  by  taking  15  parts 
by  weight  of  saltpeter  to  3  parts  of  charcoal  and  2  parts  of 
sulphur.  Find  the  number  of  pounds  of  saltpeter  used  in 
making  1  T.  of  gunpowder. 

9.  In  Ex.  8  find  the  number  of  pounds  of  charcoal  used; 
the  number  of  pounds  of  sulphur  used. 


PROPORTION  59 

Proportion.  An  expression  of  equality  between  two  ratios 
is  called  a  proportion. 

For  example,  the  ratio  $2 :  $3  is  equal  to  the  ratio  10':  15'. 
Therefore  $2:  $3  =10':  15'  is  a  proportion.  This  proportion 
is  read  "  $2  is  to  $3  as  10'  is  to  15'."  It  may,  of  course,  be 
written  simply  2  :  3  =  10  :  15,  or  |  =  i «. 

Extremes  and  Means.  The  first  and  last  terms  of  a  pro- 
portion are  called  the  extremes  \  the  second  and  third  terms 
are  called  the  means.    In  the  proportion  3^7  =  15:35,  or 

15  —  3 

3  5  — y 
if  we  multiply  both  fractions  by  7  x  35  we  see  that 

7x15  =  3x35. 

Therefore,  it  follows  that  in  any  proportion 

The  product  of  the  ineans  equals  the  product  of  the  extremes* 

If  we  let  X  stand  for  some  number  that  we  are  to  find, 

^^^^  i^  a::  29  =  26:105, 

^,        •  29  X  26 

then  X  =  — — — — 

105 

In  any  proportion  the  product  of  the  means  divided  by  either 
extreme  equals  the  other  extreme^  and  the  product  of  the  ex- 
tremes divided  by  either  mean  equals  the  other  mean. 

Illustrative  Problems.  1.  If  a::  7  =  13:21,  find  the  value 
of  X, 

As  above,  we  see  that       x  =  — — —  =  —  =  4  J. 

3 
2.  If  19.17:  21.3  =  a; :  3,  find  the  value  of  x. 

Here  we  see  that  x  =  ^-— — '■ —  = '—  =  2.7. 

/c^?  71 

7.1 


60  RATIO  AND  PROPORTION 


Exercises.    Proportion 


1.  A  student  draws  a  plan  of  the  gable  end  of  a  roof  as 
shown  below,  using  8'^  to  represent  20'.  What  length 
in  the  plan  represents  the  7 1 -foot 
''  rise "  ?  How  can  the  student  find 
the  length  of  the  slope  of  the  roof 
from  the  plan  ?  ^  ^ ^'      v^y 

2.  The  instructor  says  that  the  plan  in  Ex.  1  is  not  as 
convenient  as  a  plan  showing  half  of  the  gable  in  which 
the  10-foot  ''run"  is  represented  by  10'^  Draw  such  a  plan 
and  measure  the  slope.    How  long  is  the  slope  of  the  roof? 

3.  In  solving  Ex.  1  a  carpenter  would  use  his  square.  On 
the  tongue,  or  short  arm,  he  would  take  a  point  7V'  from  the 
corner ;  then  on  the  blade,  or  long  arm,  he 
would  take  a  point  10'^  from  the  corner. 
He  would  then  measure  the  distance  be- 
tween these  points,  and  say  that  the  num- 


A 


ber  of  inches  in  this  distance  is  the  same     ' 

as  the  number  of  feet  in  the  slope  of  the  roof.    Draw  the 
figure  to  scale  and  write  out  the  reason  involved. 

4.  By  means  of  a  pantograph  a  student  enlarges  the  floor 
plan  for  a  house  in  the  ratio  of  7:4.  If  the  dining  room  in 
the  original  plan  measures  2|"  by  3'^,  what  are  the  dimen- 
sions in  the  enlarged  plan? 

The  pantograph  is  extensively  used  in  enlarging  or  reducing  plans 
or  designs.  The  bars  of  the  instrument  here  shown  are  pivoted  at  B, 
C,  E,  and  T,  and  the  point  A  is  fixed.    As  the  ^ 

tracing  point  T  is  moved  over  the  outline  of  the        ^^%:M.^f^^^^^^^^^^^S^ 
design  which  is  being  enlarged,  the  pencil  at  P      ^^^^^^^^^i^^:::^^^^^^^^^ 
draws  the  enlargement  similar  to  the  design.  <^        ,_r>rs^ 

5.  How  far  is  it  around  a  piece  of  land  represented  by  a 
rectangle  14^'  by  18"  on  a  map  to  the  scale  of  1"  =  0.8  mi.? 


I 


PROPOFvTION  61 

6.  If  8  T.  of  coal  cost  170.96,  how  much  will  48i  T.  cost  ? 

Since  the  ratio  of  the  costs  is  equal  to  the  ratio  of  the  number  of 
tons  in  each  case,  x  :  70.96  =  48  i  :  8. 

7.  If  18  hammers  cost  1 24,  what  is  the  cost  of  28  hammers  ? 

8.  If  1000'  of  double-braided  stranded  wire  cost  150.67, 
how  much  will  850'  cost  ? 

9.  In  Ex.  8  find  to  the  nearest  foot  the  amount  of  wire 
which  you  could  buy  for  |69.50. 

10.  An  iron  casting  weighing  425  lb.  costs  $18.75.  At  this 
rate  what  is  the  cost  of  a  similar  casting  weighing  3791b.? 

11.  Two  joists  6''  wide  are  fitted  together  at  right  angles, 
as  here  shown.  The  distance  from  ^  to  ^  is  8',  that  from 
A  to  C  is  6\  and  that  from  B  to  (7  is  10'.  In  fitting  another 
joist  along  the  dotted  line  BC  the  carpenter 

has  to  saw  off  the  ends  of  the  first  joists  on 
the  slant.  Find  the  length  of  the  slanting 
cut  across  the  upright  piece ;  across  the 
horizontal  piece. 

Express  the  results  in  each  case  in  inches. 

12.  If  a  certain  type  of  engine  requires  ^  pt.  of  kerosene 
per  horse  power  per  hour,  how  much  kerosene  will  be  re- 
quired per  hour  for  a  10  H.  P.  engine  of  this  type  ?  How 
much  kerosene  will  be  required  for  this  engine  for  6  hr.  ? 

13.  If  a  boiler  evaporates  9  lb.  of  water  per  hour  per  pound 
of  coal  used,  how  many  pounds  of  water  will  it  evaporate  in 
8  hr.  if  7  T.  of  coal  are  used  ? 

Such  a  problem  may  be  solved  by  two  proportions,  or  it  may  be 
solved  without  using  any  proportion  whatever. 

14.  If  the  weight  of  a  certain  quality  of  sheet  steel  is 
487.7  lb.  per  cubic  foot,  what  is  the  weight  of  a  plate  of 
this  steel  which  is  ^^g- "  thick  and  has  3  sq.  ft.   of  surface  ? 


62 


RATIO  AND  PROPORTION 


Inversely  Proportional.    Sometimes  the  ratio  of  the  elements' 
of  one  figure  is  not  equal  to  the  ratio  of  the  corresponding 
elements  of  another  figure,  but  is  equal  to  the  inverse  of  that 
ratio.    The  elements  are  then  said  to  be  inversely  proportional. 

For  example,  the  figure  below  represents  a  large  wheel  on 
a  driving  shaft  connected  by  a  belt  with  a  smaller  wheel. 


Since  the  two  wheels  are  connected  by  the  belt,  the  ratio  of 
their  speeds  must  be  equal  to  some  ratio  of  their  diameters. 
But  since  the  belt  travels  at  the  same  rate  as  a  point  on  the 
circumference  of  the  larger  wheel,  the  number  of  revolutions 
per  minute  (R.P.M.)  of  the  smaller  wheel  must  be  greater 
than  that  of  the  larger  wheel.  Therefore,  instead  of  having 
a  proportion  in  which  the  speed  of  the  smaller  wheel  is  to 
the  speed  of  the  larger  wheel  as  the  smaller  diameter  is  to  the 
larger  diameter,  the  latter  ratio  is  inverted. 

Technically  such  wheels  for  transmitting  power  are  called  pulleys. 

Illustrative  Problem.    Using  the  data  given  in  the  above 
figure,  find  the  speed  of  the  5-inch  driven  pulley. 

Since  the  speeds  of  the  pulleys  are  inversely  proportional  to  the 
diameters,  we  have  the  proportion 

a::  200  =  10:  5, 
2 

200  X  ;^ 


from  which 


^ 


=  400. 


The  speed  of  the  driven  pulley  is  therefore  400  R.P.M. 
In  problems  of  this  type,  if  the  result  is  not  a  whole  number,  the 
nearest  whole  number  should  be  taken. 


INVERSE  PROPORTION  63 

Exercise.   Inverse  Proportion 

1.  A  24-inch  pulley  fixed  to  a  line  shaft  which  makes 
500  R.P.M.  is  belted  to  an  8-inch  pulky.  Find  the  number 
of  R.P.M.  of  the  smaller  pulley. 

2.  A  driving  pulley  has  a  diameter  of  15"  and  its  speed  is 
180  R.P.M.    Find  the  speed  of  a  6-inch  driven  pulley. 

3.  A  ch'cular  saw  with  a  6-inch  pulley  is  to  be  driven  at 
1000  R.P.M.  from  a  line  shaft  which  makes  240  R.P.M. 
What  should  be  the  diameter  of  the  driving  pulley  on  the 
line  shaft  in  order  to  obtain  this  speed? 

4.  The  diameters  of  the  steps  of  a  cone  pulley  are  4^", 
6i",  7|'',  and  9|''  respectively,  and  the  pulley  is  driven  by  a 
similar  cone  pulley  with  the  steps  arranged  in  reverse  order 
on  a  shaft  making  210  R.P.M.  Beginning  with  the  belt  on 
the  largest  step  of  the  driving  pulley,  find  the  speed  of  the 
driven  pulley  for  each  position  of  the  belt. 

A  cone  pulley  is  illustrated  in  the  blueprint  on  page  3. 

5.  A  driving  gear  with  36  teeth  meshes  with  a  driven 
gear  with  20  teeth,  as  shown  in  the  figure.  The  driving  gear 
makes  60  R.P.M.    Find  the  number  of  ,         ^        ^    , 

y^  ^^^  36  Teeth 

R.  P.  M.  made  by  the  driven  gear. 


A  gear  is  a  wheel  with  teeth  cut  on  the  rim 

to  prevent  slipping  in  transmitting  power. 

In  the  figure  the  individual  teeth  are  not  ,.  ^/    20T    th- 

shown,  the  words  "  86  Teeth  "  and  "  20  Teeth " 

indicating  that  the  wheels  are  gears.   The  rule  for  gears  is  similar  to 

that  for  pulleys,  and,  expressed  as  a  proportion,  is  as  follows : 
number  of  R.  P.  M.  of  driven  gear  _  number  of  teeth  on  driving  gear 
number  of  R.P.M.  of  driving  gear      number  of  teeth  on  driven  gear 

6.  A  gear  having  76  teeth  meshes  with  one  having  30  teeth. 
At  what  speed  should  the  smaller  gear  be  driven  so  that  the 
larger  gear  will  make  115  R.P,M.  ? 


64 


RATIO  AND  PROPORTION 


Pulley  Train.  A  series  of  pulleys  connected  by  belting, 
as  here  shown,  the  power  coming  from  one  of  the  pulleys, 
is  called  a  i)ulley  train. 

In  the  figure  the  10-inch 
pulley  and  the  8-inch  pulley 
are  fixed  to  the  same  shaft  and 
consequently  they  revolve  at 
the  s^me  speed. 

There  is  a   simple  rule 
which  covers  the  relation 
of  the  diameters  and  the  R.P.M.  of  the  pulleys  and  which 
we  express  in  the  form  of  a  proportion  as  follows: 

R.P.M.   of   last  driven  pulley 
R.P.M.  of  lirst  driving  pulley 

_  product  of  diameters  of  all  driving  pulleys 
product  of  diameters  of  all  driven  pulleys 

For  example,  in  the  pulley  train  shown  above  find  the 
speed  of  the  6-incli  pulley. 

3       5 


9:-; 


X 

150 


12  X  10 

8  X  « 


whence     x  = 


^x^ 


=  375. 


Hence  thd  speed  of  the  6-inch  pulley  is  375  R.P.M. 

Gear  Train.  A  series  of  gears  running  together,  the  power 
coming  from  one  of  the  gears,  is  called  a  gear  traiii. 

The  rule  for  a  gear  train  is  similar  to  the  rule  for  a  pulley 
train,  and,  expressed  as  a  proportion,  is  as  follows  : 

R.P.M.  of  last  driven  gear 
R.P.M.  of  first  driving  gear 

product  of  number  of  teeth  of  driving  gears 
product  of  number  of  teeth  of  driven  gears 


PULLEYS  AND  GEARS 


65 


Exercises.   Pulley  Trains  and  Gear  Trains 

1.  The  figure  below  shows  the  belt  connections  of  a  lathe, 
the  power  from  the  motor  being  transmitted  through  the  pul- 
ley train  as  shown.   Fmcl  the  number  of  R.P.  jNI.  of  the  lathe. 

Motor  750  R.P.M 


2.  In  Ex.  1,  if  a  new  motor  having  a  6-inch  pulley  and  a 
speed  of  325  R.P.M.  were  put  in,  what  size  of  driven  pulley 
to  the  nearest  ^  would  be  needed  on  the  line  shaft  in  order 
to  run  the  lathe  at  the  same  speed  ? 

3.  In  Ex.  2  what  size  of  pulley  would  be  needed  in  order 
to  run  the  lathe  at  a  speed  of  350  R.P.M.? 

4.  In  the  gear  train  here  shown  the  driving  shaft  has  a 
speed    of    50    R.P.M.     Find   the 
speed  of  the  last  driven  gear. 

The  symbol  60  T  means  that  the  gear 
has  60  teeth.  This  conventional  symbol 
will  hereafter  be  used  in  connection  with 
gears  and  gear  trains. 

5.  In  Ex.  4  suppose  that  the  72-T  gear  were  the  driving 
gear  and  that  its  shaft  made  100  R.P.M.,  what  would  then 
be  the  speed  of  the  60-T  gear  ? 


72  T 


66  RATIO  AND  PROPORTION 

6.  The  change  gears  in  the  blueprint  on  page  67  are  used 
on  a  lathe  when  cutting  screw  threads.  From  the  data  on  the 
drawing  find  the  speed  of  the  lead  screw. 

7.  In  Ex.  6  find  the  number  of  teeth  that  the  driven  gear 
on  the  compound  should  have  in  order  that  the  speed  of  the 
lead  screw  may  be  doubled. 

Express  the  result  in  such  a  problem  as  the  nearest  whole  number. 

8.  In  the  gear  train  shown  in  the  blueprint  find  the  speed 
of  the  52-T  gear. 

9.  If  the  52-T  gear  in  the  gear  train  were  the  driving 
gear,  and  if  its  speed  were  125  R. P.M.,  what  would  be  the 
speed  of  the  28-T  gear? 

10.  In  the  lathe  gear  box  shown  in  the  blueprint,  gears  A 
and  B  can  be  brought  into  engagement  with  any  of  the  gears 
on  the  lead  screw.  Gear  B  is  an  idler,  or  an  intermediate 
gear,  and  has  no  effect  upon  the  speed  of  the  driven  gear  on 
the  lead  screw.  Find  the  speed  of  the  lead  screw  when  gears 
A  and  B  engage,  in  turn,  each  of  the  gears  on  the  lead  screw, 
beginning  with  the  smallest. 

11.  In  the  planer  gears  shown  in  the  blueprint  find  the 
speed  of  the  last  driven  gear. 

12.  What  would  be  tlie  result  m  Ex.  11  if  the  driving  shaft 
made  110  R. P.M.? 

13.  In  Ex.  11  what  would  be  the  result  if  the  55-T  gear 
were  changed  to  a  60-T  gear  ? 

14.  In  Ex.  11  what  would  be  the  result  if  we  should 
replace  the  62-T  gear  with  a  50-T  gear  ? 

15.  The  line  shaft  in  a  machine  shop  runs  at  200  R.P.M. 
A  grinder  with  a  pulley  6"  in  diameter  should  run  at 
1800  R.P.M.  In  order  to  obtain  this  speed,  what  should 
be  the  diameter  of  the  pulley  placed  on  the  line  shaft? 


PULLEYS  AND  GEAES 


67 


->Stud    \ 

35M.P.M.\ 


.npoutid 


'"''>1a.   V'" 


,Qad  Screu 


CHANGE  GEAF(S 


GEAf^  TF\AIN 


KiriB==l 

ililll 

■■n  8^ 

all 

Lead   Screw 


Driving  iShaft 


LATHE  GEARBOX 


Driving  fShaftl 


PLANEF(  GEA^S 


68 


RATIO  AND  PROPORTION 


,26T 


-76  R.P.M 


16.  In  this  gear  train  find  the  speed  of  the  gear  marked  C. 

It  should  be  noticed  that  gear  B  in 
this  train  is  an  idler,  or  intermediate 
gear.  Such  a  gear  was  shown  in  the  lathe 
gear  box  on  page  67  and  is  frequently 
used  where  it  is  desired  to  transmit  power 
between  two  points,  such  as  ^1  and  C,  which 
are  too  far  apart  to  use  only  two  gears  of 
convenient  size.  An  intermediate  gear 
does  not  affect  the  speed  of  the  driven  gear. 

17.  In  the  gear  train  below  find  the  speed  of  the  64-T  gear. 

'64  T 


122  R.P.M. 


18.  In  Ex.  17,  if  the  64-T  gear  were  the  driving  gear  and 
made  100  R.P.M.,  what  would  then  be  the  number  of  R.P.M. 
of  the  16-Tgear? 

19.  In  the  blueprint  on  page  69,  which  shows  how  power 
is  transmitted  by  belts,  find  the  speed  of  the  surfacer. 

20.  Find  the  number  of  R.P.M.  of  the  grindstone. 

21.  Find  the  speed  of  the  circular  saw. 

22.  Find  the  number  of  R.P.M.  of  the  band  saw. 

23.  What  size  of  driven  pulley  to  the  nearest  y  should  be 
placed  on  the  grindstone  to  increase  its  speed  to  60  R.P.M.  ? 
to  increase  its  speed  to  75  R.P.M.? 

24.  What  would  be  the  speed  of  the  band  saw  if  the  18-inch 
pulley  on  the  shaft  which  drives  the  grindstone  were  placed 
so  as  to  drive  the  band  saw  direct  ? 


PULLEYS  AND  GEAKS 


69 


70  RATIO  AND  PROPORTION 

Exercises.  Review 

1.  If  1'^  on  a  map  represents  375  mi.,  what  is  the  distance 
between  two  places  which  are  2i"  apart  on  the  map  ? 

2.  If  2.8  bbl.  of  lime  are  required  for  75  sq.  yd.  of  plaster- 
ing, how  many  barrels  are  needed  for  675  sq.  yd.? 

3.  A  stretch  of  railroad  track  runs  685',  with  a  uniform 
grade  of  8|"  per  100'.  What  is  the  difference  in  level  be- 
tween the  bottom  and  the  top  of  the  grade  ? 

4.  Gun  metal  is  composed  of  1  part  of  tin  to  5  J  parts  of 
copper  by  weight.  How  many  pounds  of  tin  must  be  added 
to  420  lb.  12  oz.  of  copper  to  make  gun  metal? 

5.  How  many  pounds  of  tin  are  there  in  464  lb.  12  oz.  of 
gun  metal  such  as  that  described  in  Ex.  4  ? 

6.  If  35  men  in  16  da.  can  complete  half  of  an  excavation, 
how  long  would  it  take  to  complete  the  other  half  if  five  more 
men  were  added  to  the  working  force  ? 

7.  A  14-inch  pulley  fixed  to  a  line  shaft,  wdiich  runs  at 
150  R.P.  M.,  is  belted  to  a  12-inch  pulley  on  a  countershaft. 
Find  the  number  of  R.P.M.  of  the  countershaft. 

8.  A  grindstone  with  a  28-inch  pulley  is  to  be  driven 
at  50  R.P.M.  from  the  countershaft  in  Ex.  7.  What  size 
of  driving  pulley  should  be  placed  on  the  countershaft  to 
obtain  this  speed  ? 

9.  A  6-inch  pulley  on  a  countershaft  drives  a  22-inch 
pulley  on  a  hacksaw.  What  should  be  the  speed  of  the 
countershaft  to  drive  the  hacksaw  at  33  R.P.M.? 

10.  What  size  of  pulley  should  be  placed  on  the  counter- 
shaft in  Ex.  9  to  increase  the  speed  of  the  hacksaw  33i%  ? 

11.  A  grinder  which  should  have  a  speed  of  480  R.P.M.  is 
to  be  run  by  a  19-inch  pulley  on  a  shaft  making  180  R.P.M. 
What  size  of  pulley  should  be  placed  on  the  grinder  ? 


CHAPTER  III 

MENSURATION 

Common  Measures.  Industry  in  general  makes  use  of  a 
relatively  small  number  of  the  measures  which  are  usually 
taught  in  the  schools. 

In  measuring  lengths  the  inch,  foot,  and  yard  are  the 
most  common  units.  While  the  abbreviations /it.  and  in.  are 
used  for  feet  and  inches  respectively,  the  symbols  '  and  '^  are 
more  common  in  the  shop. 

In  practical  measuring,  fractional  parts  of  an  inch  are 
usually  expressed  with  denominators  2,  4,  8,  16,  32,  or  64 ; 
less  often  with  denominators  3,  6,  12,  24,  or  48 ;  and  with 
growing  frequency  as  decimals. 

The  common  units  used  in  measuring  areas  are  given  on 
page  72,  and  those  used  in  measuring  solids  on  page  84. 

Liquid  and  dry  measures  are  used  in  industry,  but  many 
substances  that  were  formerly  measured  by  the  quart,  gallon, 
or  bushel  are  now  measured  by  weight. 

In  measuring  weight,  the  ounce,  pound,  and  ton  of  2000  lb. 
are  used,  although  the  long  ton  of  2240  lb.  is  still  found. 

The  metric  measures  are  considered  on  pages  122-130. 

The  tables  on  pages  193-194  should  be  consulted  if  necessary. 

In  finished  machine  work,  results  are  usually  carried  to 
the  nearest  0.001'' ;  in  plumbing,  carpentry,  and  sheet-metal 
work,  to  the  nearest  J^'';  in  weights,  to  the  nearest  hundredth 
of  the  unit  employed ;  and  in  speeds  of  pulleys,  gears,  and 
machines,  to  the  nearest  unit. 

71 


72 


MENSURATION 


Square  Measure.  In  measuring  areas  the  square  inch, 
square  foot,  and  square  yard,  with  their  decimal  subdivisions 
or  with  the  simplest  common  fractions,  are  used. 

A  square  inch  (sq.  in.)  is  the  area  of  a  square  that  is  one 
inch  (1'^)  on  a  side  ;  a  square  foot  (sq.  ft.)  is  the  area  of  a 
square  that  is  one  foot  (!')  on  a  side  ;  and  a  square  yard  (sq.yd.) 
is  the  area  of  a  square  that  is  one  yard  (1  yd.)  on  a  side. 


■ 

J 

.8 

(Z| 

-1 

llnchir 

1  Yard 


k-1  f  t.- 


The  left-hand  figure  is  drawn  to  the  scale  y^^,  and  consequently  the 
square  which  represents  1  sq.  ft.  is  0.1',  or  1.2",  on  a  side.  It  will  be  seen 
that  there  are  144  sq.  in.  in  1  sq.  ft.  The  right-hand  figure  is  drawn  to 
the  scale  y^^'  ^^^  the  square  which  represents  1  sq.  yd.  is  ^'^  of  a  yard, 
or  1.2",  on  a  side.   It  will  be  seen  that  there  are  9  sq.  ft.  in  1  sq.  yd. 

Area  of  a  Rectangle.  For  example,  if  the  floor  of  a  rec- 
tangular hall,  which  is  10'  long  and  4'  wide,  is  made  of  marble 
squares,  1'  on  a  side,  as  here  shown,  there  are  10  squares  in 
each    row,    and    there    are    4 


iiiiirS^^^ 


rows  of  squares.     Since  there 
are  4x10  squares,  we  have 

Area  =  4  x  10  sq.  ft. 

=  40  sq.  ft. 

To  be  more  precise  we  might  write  4  x  10  =  40,  the  number  of  square 
feet,  but  for  brevity  the  abbreviation  sq.fi.  is  commonly  used  as  shown. 

The  area  of  a  rectangle  is  the  product  of  the  base  and  height. 


SQUARE  MEASURE  73 

Exercises.    Square  Measure 

1.  Find  the  floor  area  of  a  room  28'  6"  long  and  18'  wide. 

2.  Find  the  cross-section  area  of  a  square  beam  3|"  on  a 
side.  Verify  the  result  by  drawing  a  3i-inch  square  and 
ruling  it  off  into  |-inch  squares. 

3.  A  room  16'  by  19'  is  9|'  high.  Find  the  total  area  of  the 
four  walls,  the  floor,  and  the  ceiling. 

No  allowance  for  doors  and  windows  is  to  be  made  unless  specified. 

4.  A  garden  48'  by  66'  contains  a  3-foot  walk  laid  inside 
the  garden  along  the  four  sides.  The  mid  points  of  the  long 
sides  are  joined  by  a  2-foot  path.  Find  the  area  left  for 
cultivation  and  draw  a  plan  to  any  convenient  scale. 

5.  On  the  floor  of  a  room  36'  long  and  28'  wide  a  border 
2'  wide  is  to  be  painted.  Find  the  cost  of  painting  the  border 
at  600  per  square  yard. 

6.  At  260  per  square  foot  find  the  cost  of  a  concrete 
walk  6'  wide  round  the  outside  of  a  garden  66'  by  84'. 

7.  Draw  a  plan  to  any  convenient  scale  of  the  basement 
of  a  house,  the  walls  of  which  are  laid  out  as  follows; 
Starting  at  the  southeast  corner  the  wall  runs  north  22', 
then  west  12',  north  14',  west  16',  south  36',  and  east  to  the 
starting  point  28',  all  dimensions  being  inside  measurements. 
Find  the  cost  of  cementing  the  floor  at  250  per  square  foot. 

8.  A  rectangle  is  6|-"  long  and  41"  wide.  Find  the  area 
correct  to  the  nearest  ^  sq.  hi. 

9.  A  workman  measures  the  side  of  a  square  and  finds  it 
to  be  5.24",  but  the  last  figure  is  uncertain,  being  possibly 
either  3  or  5.  If  he  uses  5.24"  for  finding  the  area  of  the 
square,  what  is  the  greatest  possible  error  that  can  arise  ? 

First  find  the  values  of  5.24  x  5.24,  5.23  x  5.23,  and  5.25  x  5.25. 


74  MENSURATION 

Formula.  In  all  kinds  of  applied  mathematics  there  is 
need  for  symbols.  Thus  we  use  +  for  ''  plus,"  —  for  "minus," 
X  for  "  times,"  -f-  for  "  divided  by,"  and  V  for  "  the  square 
root  of."  This  form  of  mathematical  shorthand  is  still 
further  extended  when  we  come  to  the  rules  for  jneas- 
uring.  Instead,  for  example,  of  writing  "  The  area  of  a  rec- 
tangle is  the  product  of  the  base  and  height,"  we  simply 
write  the  formida 

A  =  bh, 

using  the  initial  letters  for  ''  area,"  ''  base,"  and  "  height," 
and  understanding  that  when  two  letters  are  written  side  by 
side  their  product  is  to  be  taken. 

In  practical  mathematics  we  have  no  time  for  long  rules 
when  we  can  more  easily  express  these  rules  by  formulas. 

Either  capital  letters  or  small  letters  may  be  used  in  a  formula. 

Use  of  the  Formula.  For  example,  if  the  base  of  a  rectangle 
is  9''  and  the  height  is  7|",  find  the  area. 

Since  the  dimensions  were  given  in  inches,  the  area  is  67^  sq.  in. 

Evaluating.  In  using  such  a  formula  sls  A  =  bh  the  student 
will  often  be  asked  to  find  the  value  of  A  when  h  and  h 
have  certain  given  values.  It  is  convenient  to  have  a  single 
word  to  use  for  the  expression  "find  the  value  of,"  and  we 
use  the  word  evaluate  for  this  purpose. 

For  example,  to  evaluate  bh  for  the  values  b  =  6  and  7i  =  4 
we  have  &7i  =  6  x  4  =  24.  That  is,  bh  =  24  for  these  values 
of  b  and  h. 

Similarly,  we  evaluate  a^  for  a  =  7  by  writing  7  for  a, 
whence  ^2  =  7x7=49. 

The  expression  a^,  read  "a  square,"  means  aa,  or  a  x  a.  Similarly, 
a^,  read  "  a  cube,"  means  aaa,  or  a  x  a  x  a. 


FORMULAS  76 

Exercises.   Formulas 

1.  The  area  of  a  square  of  side  s  is  s^.  The  formula  for 
the  area  may  therefore  be  written  A  =  8^.  Evaluate  this 
formula  for  A  when  «=  2^". 

2.  From  a  steel  beam  weighing  b  pounds  there  hangs  a 
block  and  tackle  weighing  t  pounds,  and  by  this  a  load  of 
w  pounds  is  being  lifted.  Write  a  formula  for  X,  the  total 
resulting  load  on  the  beam,  including  the  weight  of  the  beam. 

3.  Evaluate  the  formula  found  in  Ex.  2  for  X,  given  that 
b  =  750,  t  =  325,  and  w  =  1675. 

4.  The  area  of  a  rectangle  is  shown  on  page  74  to  be 
expressed  by  the  formula  A=  bh.  Evaluate  this  formula  for 
A  when  J  =  7.5  and  h  =  3.4. 

5.  Evaluate  the  formula  of  Ex.  4  for  A  when  J  =  10.2  and 
h  =  7.3.  If  b  and  h  are  dimensions  in  inches,  what  is  the 
area  of  the  rectangle  ? 

6.  It  will  be  shown  on  page  76  that  the  area  of  a  triangle 
of  base  b  and  height  h  is  expressed  by  the  formula  A  =  i  bh. 
Evaluate  this  formula  for  A  when  5  =  3|  and  A  =  11. 

7.  The  area  of  a  circle  is  expressed  by  the  formula  ^  =  -2y2-  y^. 
Find  the  value  of  A  when  r,  the  radius,  is  35  units. 

If  the  unit  is  1",  the  area  will  be  in  square  inches ;  if  the  unit  is  1', 
the  area  will  be  in  square  feet, 

8.  An  odd  number  is  indicated  by  the  expression  2^1+1, 
where  n  is  0  or  any  whole  number.  In  this  expression  give 
to  n  all  the  various  values  from  0  to  10  and  evaluate  the 
expression  for  each  of  these  values,  thus  obtaining  all  the  odd 
numbers  from  1  to  21. 

While  not  a  practical  problem,  this  shows  the  general  nature  of  a 
formula  and  gives  an  idea  of  the  meaning  of  an  algebraic  expression, 
p 


76 


MENSURATION 


Area  of  a  Parallelogram.  If  from  any  parallelogram,  like 
J  BCD  in  the  left-hand  figure  below,  we  cut  off  the  shaded 
triangle  T  by  a  line  perpendicular  to  DC  and  place  this 
triangle  at  the  other  end  of  the  parallelogram,  as  shown  in 
the  figure  at  the  right,  the  resulting  figure  is  a  rectangle. 


D 


That  is,  the  area  of  a  parallelogram  is  equal  to  the  area  of 
a  rectangle  of  the  same  base  and  the  same  height.  Since 
the  formula  for  the  area  of  a  rectangle  is  ^  =  bh, 

The  area  of  a  parallelogram  is  the  product  of  the  base  and 
height. 

This  rule  may  be  conveniently  expressed  by  the  formula 

A  =  bh. 

Area  of  a  Triangle.  If  we  draw  the  diagonal  ^C  in  the 
parallelogram  ABCD   below,    we    divide    the   parallelogram 

into  two  equal  obtuse  triangles.  jy (^ 

The  diagonal  DB  would  give  two 
equal   acute   triangles,    and   the 
diagonal   of   a   rectangle   would     ^. 
give  two  equal  right  triangles. 

Therefore,  since  any  triangle  is  half  of  a  parallelogram  of 
the  same  base  and  the  same  height,  we  see  that 

The  area  of  a  triangle  is  half  the  product  of  the  base  and 
height 

This  rule  may  be  conveniently  expressed  by  the  formula 

A  =  \bh. 


PARALLELOGRAM  AND  TRIANGLE 


77 


Exercises.    Parallelograms  and  Triangles 

Draw  the  following  parallelograms  to  scale  and  find  the  area 
of  the  origiyial  parallelogram  and  of  the  drawing  in  each  case : 

1.  Base,  40^';  other  side,  30'^  height,  16'^;  scale  \. 

2.  Base,  16'';  other  side,  9'';  height,  8'';  scale  ^. 

3.  Base,  4.5'';  other  side,  9.6";  height,  3.75";  scale  1. 

4.  The  base  of  a  triangle  is  5.7"  and  the  height  of  the 
triangle  is  3.4".  Find  the  area  of  the  triangle  correct  to 
the   nearest  0.1  sq.  in. 

5.  Find  the  cost  of  plastering  the  walls  and  ceiling  of  a 
room  28'  9"  by  36'  6"  and  11'  high  at  68  (f  per  square  yard, 
deducting  20%  of  the  wall  area  for  doors  and  windows. 

6.  What  fraction  of  a  square  yard  of  bunting  is  there  in 
a  triangular  pennant  which  has  a  width  of  27"  measured  along 
the  flagstaff  and  a  length  of  2  yd.  ? 

7.  The  span  AB  of  a  roof  is  40', 
the  rise  MC  is  15',  the  slope  ^C  is  25', 
and  the  length  BE  is  60'.  Fmd  the 
area  of  each  gable  end  and  also  the 
area  of  the  roof. 

In  this  problem  do  not  consider  any  overhang  of  the  eaves. 

8.  In  this  figure  ABCD  represents  a  6-inch  square,  E^  E,  G, 
and  Jf  being  the  mid  points  of  the  sides.  In  the  square  AEOH, 
AP=QE=EE=SO=  ...  =WA  =  \AE. 
Find  the  area  of  each  of  the  small  tri- 
angles such  as  APW  and  also  of  the 
octagon  PQRSTUVW. 

The  dots  (  •  •  •  )  mean  "  and  so  on  "  and 
in  this  particular  case  they  take  the  place  of 
OT=UH=HV. 

An  octagon  is  a  figure  of  eight  sides. 


78  MENSURATION 

Area  of  a  Trapezoid.  If  a  trapezoid  T  has  its  duplicate  cut 
from  paper  and  turned  over  and  fitted  to  it,  as  D  in  this 

figure,  the  two  together  form        , r -. 

a  parallelogram.     How  does      /  T         \  d 

the  area  of  the  whole  paral-     ^ ' 

lelogram  compare  with  the  area  of  the  trapezoid  T?  How 
does  the  base  of  the  parallelogram  compare  with  the  sum  of 
the  upper  and  lower  bases  of  the  trapezoid  ?  How  do  you 
find  the  area  of  the  parallelogram  ?  Then  how  do  you  find 
the  area  of  the  trapezoid  ?  j)  r 

If  from  the  trapezoid  ABCD^  shown  in 
this  figure,  the  shaded  portion  is  cut  off 
and  is  fitted  into  the  space  marked  by  the      ^  ^ 

dotted  lines,  what  kind  of  figure  is  formed  ?  How  is  the 
area  of  the  resulting  figure  found  ? 

From  these  illustrations  we  infer  the  following: : 

The  area  of  a  trapezoid  is  the  j)^oduct  of  one  half  the  height 
and  the  sum  of  the  j»:>ar6i?ZeZ  sides. 

This  rule  may  be  conveniently  expressed  by  the  formula 

where  A  stands  for  the  area,  h  for  the  height,  B  for  the 
lower  base,  and  h  for  the  upper  base. 

The  parentheses  show  that  B  and  b  are  to  be  added  before  the  sum 
is  multiplied  by  \  h.    For  example,  if  A  =  4,  5  =  7,  and  b  =  o,  we  have 

A  =  lh(B-\-  b) 
=  1  X  4  X  (7  +  5) 
=  2  X  12  =  24. 

Since  all  ordinary  rectilinear  figures,  that  is,  figures  formed  by  straight 
lines,  are  made  up  of  parallelograms  of  some  kind  (including  squares 
and  rectangles),  triangles,  or  trapezoids,  we  have  now  found  how  to 
measure  the  area  of  a  rectilinear  figure  of  any  shape. 


AREAS  79 

Exercises.    Areas 

1.  Find  the  area  of  a  piece  of  ground  in  the  form  of  a 
trapezoid,  the  parallel  sides  being  63J'  and  37|'  and  the 
perpendicular  distance  between  these  parallel  sides  being  24'. 

2.  If  the  area  of  a  trapezoid  is  396  sq.  in.  and  the  bases 
e  19'^  and  21^'  respectively,  what  is  the  height  ? 

3.  The  bases  of  a  trapezoid  are  S^"  and  6|''  respectively 
and  the  height  is  ^jq''.  Find  the  area  of  the  trapezoid  correct 
to  the  nearest  |  sq.  m. 

4.  Draw  to  any  convenient  scale  a  plane  rectilinear  figure 
with  seven  sides  (a  heptagon)  and  show  that  its  area  can  be 
found  by  cutting  the  figure  into  smaller  figures,  the  areas  of 
which  can  be  found  by  the  formulas  already  given. 

5.  A  floor  is  paved  with  six-sided  tiles,  as  here  shown.  In  the 
picture  the  tiles  have  been  divided  by  dotted  lines  to  suggest 
a  method  of  measuring  them.  What 
measurements  would  you  take  to  find 
the  area  of  each  tile  ?  What  other  divi- 
sions of  the  tiles  can  you  suggest  for 
convenience  in  finding  this  area? 

6.  The  sides  of  a  rectangle  are  given  as  3'^  and  3.8^', 
where  the  width  is  exact  and  the  length  is  correct  to  the 
nearest  0.1'^  The  length  therefore  lies  between  3.75"  and 
3.85'^  Draw  the  figure  as  accurately  as  you  can  for  each  of 
these  lengths  and  show  that  the  greatest  possible  error  arising 
from  taking  3.8"  as  the  length  is  0.15  sq.  in. 

7.  The  base  of  a  triangle  is  given  as  5.6"  exactly,  and 
the  height  is  measured  as  3.2"  correct  to  the  nearest  0.1". 
Between  what  limits  does  the  area  of  the  triangle  lie? 

8.  How  many  sheets  of  tin  18"  by  24"  are  needed  to 
cover  a  roof  75'  by  120',  allowing  20  sheets  for  waste? 


80 


MENSURATION 


Estimates  of  Area.  If  a  figure  is  drawn  on  squared  paper, 
the  area  inclosed  may  be  estimated  approximately  by  count- 
ing the  squares.  In  counting  the  squares  cut  by  the  outline  of 
the  figure  exclude  any  square  which  does  not  lie  at  least  half 
within  the  figure,  count  as  a  half  square  one  that  is  practically 
evenly  divided,  and  count  as  a  full  square  one  that  lies  more 
than  half  within  the  outline. 

For  example,  in  this  figure  if  each  square  repre- 
sents 1  sq.  in.,  the  area  inclosed  by  the  curve  is 
approximately  33  sq.  in.,  as  there  are  approximately 
33  squares  inclosed. 

In  practical  work  this  method  is  commonly  used 
even  with  rectilinear  figures,  the  areas  of  which 
can  be  computed  by  the  methods  already  given. 


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s^    y 

Exercises.    Estimates  of  Areas 

1.  The  squared  paper  shown  on  page  81  is  ruled  with 
ten  lines  to  the  inch,  every  tenth  line  being  heavy.  The 
figures  are  drawn  to  the  scale  J^.  Find  the  approximate 
area  of  figure  I  by  counting  the  squares. 

2.  Find  the  area  of  figure  I  by  dividing  it  into  two  trape- 
zoids and  finding  the  area  of  each  by  the  formula  on  page  78. 

By  counting  the  squares  and  hy  dividiiig  the  figures  into 
parts,  the  areas  of  which  can  he  computed  hy  the  formulas,  fijid 
the  area  of  each  of  the  following : 

3.  Figure  II.    5.  Figure  IV.  7.  Figure  VI.      9.  Figure  VIII. 

4.  Figure  III.  6.  Figure  V.    8.  Figure  VII.  10.  Figure  XII. 

11.  As  in  Exs.  3-10,  find  the  area  of  figure  IX,  deducting 
the  area  of  the  triangle  from  the  area  of  the  whole  figure. 

12.  By  counting  squares  and  by  dividing  into  parts  in  any 
convenient  way,  find  the  area  of  figure  X ;  of  figure  XI. 


ESTIMATES  OF  AREAS 


81 


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82  MENSURATION 

Application  to  Working  Drawings.  The  blueprints  on  page  83 
represent  the  kind  of  working  drawings  from  which  the  work- 
man has  to  compute  areas.  In  computing  the  areas  in  the 
exercises  below,  the  student  may  lay  out  the  figures  on  squared 
paper,  or  he  may  apply  the  formulas  for  mensuration. 

When,  in  computing  such  areas  from  large  bhieprints,  a  dimension 
not  given  in  the  drawing  is  needed,  the  measurement  should  be  taken 
with  a  pair  of  dividers  and  then,  from  the  scale  on  the  drawing  or  by 
proportion,  the  required  dimension  should  he  found. 

Owing  to  the  small  size  to  which  these  blueprints  have  of  necessity 
been  reduced,  such  measurements  would  be  too  inaccurate,  and  the  prob- 
lems given  can  be  solved  from  the  dimensions  in  the  blueprints. 

Exercises.   Computing  Areas 

1.  The  gusset  plate  on  page  83  is  made  of  steel  weighing 
18.771b.  per  square  foot.    Fhid  the  weight  of  the  plate. 

It  will  be  desirable  in  this  case  to  lay  out  the  figure  to  the  scale  yV  or 
to  the  scale  ^  on  squared  paper  with  ten  lines  to  the  inch. 

2.  If  a  bridge  builder  has  to  chip  all  round  the  gusset 
plate  in  order  to  true  it,  what  length  does  he  chip  ? 

3.  The  angle  clamp  is  made  of  brass  0.032''  thick,  such 
brass  weighing  1.367  lb.  per  square  foot.  Find  to  the  near- 
est 0.01  oz.  the  weight  of  the  clamp. 

4.  Find  the  area  of  the  cross  section  of  the  metal  part  of, 
the  bearing  support,  as  shown  by  the  shaded  portion. 

5.  Find  the  area  of  the  side  of  the  desk. 

6.  How  much  floor  space  is  added  to  a  room  if  the  bay 
window  here    shown   is   built   out   on 

one   side    of   the  room  ?     How  much  3-2* 

floor    space  would    be    added    if    the  V  t » 

dimensions  of   the   bay  window   were  \       i 

each  made  li  times  as  great?  |<-3-2^i<     6-0- 


COMPUTING  AREAS 


83 


I 


84 


MENSUKATION 


Cubic  Measure.  In  measuring  volumes  the  cubic  inch,  cubic 
foot,  and  cubic  yard,  with  their  decimal  subdivisions  or  with 
the  simplest  common  fractions,  are  used. 

A  cubic  inch  (cu.  in.)  is  the  volume  of  a  cube  that  is  one 
inch  (1^^)  on  an  edge ;  a  cubic  foot  (cu.  ft.)  is  the  volume  of  a 
cube  that  is  one  foot  (1')  on  an  edge ;  and  a  cubic  yard  (cu.yd.) 
is  the  volume  of  a  cube  that  is  one  yard  (1  yd.)  on  an  edge. 


The  above  figures,  drawn  to  the  scale  -r^jr,  represent  respectively  from 
left  to  right  1  cu.  ft.,  1  cu.  yd.,  and  a  rectangular  solid  1  ft.  thick,  1  yd. 
long,  and  1  yd.  high.  We  see  that  the  third  solid  contains  9  cu.  ft.  and 
that  1  cu.  yd.  is  made  up  of  three  such  solids.  Hence  1  cu.  yd.  contains 
27  cu.  ft.   Similarly,  it  may  be  shown  that  1  cu.  ft.  contains  1728  cu.  in. 

Volume  of  a  Rectangular  Solid.  If  this  figure  represents 
a  rectangular  solid  5"  long,  3''  wide,  and  7'^  high,  it  is  evident 
that  in  the  column  of  cubes  shown  there  are  7  cu.  in.  It  is 
also  evident  that  on  the  base  we  can  place 
3x5  such  columns.  Hence  the  volume  is 
3  X  5  X  7  cu.  in.,  or  105  cu.  in.    Therefore 

The  volume   of  a  rectangular    solid  is  the 
product  of  the  three  dimensions. 

This  may  be  expressed  by  the  formula 
V=lwh. 

The  letters  /,  iv,  h  are  the  initials  of  "  length,"  "  width,"  and  "  height." 


r 


RECTANGULAR  SOLIDS  85 

Exercises.  Volumes 
Find  the  volumes  of  solids  of  the  following  dimensions : 

1.  23^  24^  26^      4.  27',  12^  131^        7.  36'^  28l'^  81-'^ 

2.  40'^  34'^  25'^      5.  28^  36^  14'.  8.  161'',  81",  41". 

3.  63",  48",  30".      6.  32",  23",  12f ".     9.  5f ',  2^',  If. 

10.  A  storeroom  is  66'  long,  32'  wide,  and  161'  high.  Find 
the  number  of  cubic  feet  of  space  in  the  room. 

11.  A  cellar  36'  x  32'  x  6'  is  to  be  excavated.  How  much 
will  the  excavation  cost  at  95(f  a  cubic  yard? 

Dimensions  are  often  given  in  this  form,  and  36'  x  32'  x  6'  means 
that  the  excavation  is  to  be  36'  long,  32'  wide,  and  6'  deep. 

12.  At  92^  a  cubic  yard,  how  much  will  it  cost  to  dig  a 
ditch  160'  long,  3'  wide,  and  5'  deep  ? 

13.  The  box  of  an  ordinary  farm  wagon  is  3'xlO',  and 
the  depth  is  usually  24"  or  26".  Find  the  volume  in  cubic 
feet  and  also  in  cubic  inches  for  each  of  these  depths. 

14.  The  interior  of  a  certain  freight  car  is  36'  long,  8'  4" 
wide,  and  7'  6"  high.  How  many  cubic  feet  does  it  contain  ? 
If  the  car  is  filled  with  grain  to  a  depth  of  4'  9",  what  is  the 
weight  of  the  grain  at  60  lb.  to  the  bushel,  allowing  1|  cu.  ft. 
to  the  bushel  ? 

15.  To  what  depth  must  a  rectangular  tank  6'  x  9'  8"  be 
filled  with  water  to  contain  160  cu.  ft.  of  water  ? 

16.  If  2^  cu.  ft.  of  corn  in  the  ear  produce  1  bu.  of  shelled 
corn,  how  many  bushels  of  shelled  corn  can  be  obtained  from 
a  crib  16'  x  24'  x  8'  filled  with  corn  in  the  ear  ? 

17.  A  cellar  24'  x  36'  x  7'  is  to  be  dug  for  a  house  on  a 
level  lot  62'  x  140'.  The  dirt  taken  from  the  cellar  is  to  be 
used  to  raise  the  level  of  the  whole  lot.  How  much  will  the 
level  of  the  lot  be  raised  if  the  dirt  is  evenly  distributed  ? 


MENSUEATION 


Measuring  Lumber.  A  board  foot  (1  bd.  ft.)  of  lumber 
means  a  piece  that  is  1  sq.  ft.  on  one  surface  and  V  or 
less  in  thickness.    A  board  foot  is  often  called  simply  a  foot. 


From  left  to  right  the  figures  shown  above  represent  pieces  of 
himber  that  contain  1  bd.  ft.,  4  bd.  ft.,  and  6  bd.  ft.  respectively. 

A  board  24'  long,  IC  wide,  and  1^'  or  less  thick  contains 
24  X  i|  bd.  ft,  or  20  bd.  ft.  A  beam  16'  long,  6"  wide,  and 
8''  thick  contains  16  x  ^^2  >^  ^  ^^-  ^^•'>  ^^  ^^  ^^*  ^^'  ^  fraction 
of  a  board  foot  is  counted  as  1  bd.  ft. 

We  may  express  the  number  of  board  feet  as  follows : 

where  B  is  the  number  of  board  feet,  I  the  length  m  feet, 
2V  the  width  in  inches,  and  t  the  thickness  in  inches. 

In  large  quantities  lumber  is  usually  referred  to  as  so 
many  thousand  feet.   Thus  9  M  bd.  ft.  means  9000  board  feet. 

Volume  of  a  Prism.  This  figure  represents  a  prism.  On 
each  square  foot  of  the  base  there  can  be  placed  1  cu.  ft., 
reaching  to  the  height  of  1'  m  the  prism. 
Therefore  the  shaded  part  of  the  bottom  of  the 
prism  will  contain  as  many  cubic  feet  as  there 
are  square  feet  in  the  base.  Therefore  the  whole 
prism,  which  is  4'  high,  will  contain  four  times 
as  much,  so  that  if  the  base  contains  3  sq.  ft.,  the  volume  is 
4  X  3  cu.  ft.,  or  12  cu.  ft.    That  is. 

The  volume  of  a  prism  is  the  product  of  the  base  and  height. 

This  rule  may  be  conveniently  expressed  by  the  formula 

V=bh. 


f 


MEASURING  LUMBER  87 

Exercises.    Volumes 


1'       1.  A  box  is  61"  long,  41'^  wide,  and  2^"  deep.    Write  the 
formula  for  the  volume  and  find  the  volume. 

In  the  case  of  boxes  and  other  receptacles  all  dimensions  are  inside 
measurements  except  when  otherwise  stated.  In  such  problems  estimate 
the  results  in  advance  as  a  check  on  the  accuracy  of  comjiutation. 

2.  An  excavation  is  69'  long,  45'  wide,  and  6'  4'^  deep. 
Write  a  formula  for  the  number  of  loads  of  earth  removed 
and  then  find  this  number. 

In  excavation  work  a  load  is  equivalent  to  a  cubic  yard. 

3.  A  prism  has  a  base  of  37 J  sq.  in.  and  a  height  of  16". 
Find  the  volume. 

4.  A  bin  16'  x  28'  is  filled  with  coal  to  a  depth  of  6'. 
Allowing  35  cu.  ft.  to  a  ton  of  coal,  write  a  formula  for  the 
number  of  tons  of  coal  in  the  bin  and  then  find  this  number. 

5.  A  machine  part  made  of  steel  weighing  490  lb.  per 
cubic  foot  is  cast  in  the  form  of  a  prism  which  has  a  length 
of  4'  8"  and  a  cross-section  area  of  26  sq.  m.  Find  the  volume 
and  the  weight  of  the  part. 

6.  To  floor  a  barn  136  planks,  each  12'  long,  1'  wide,  and 
2"  thick,  were  used.  From  the  formula  for  board  measure 
find  the  amount  of  lumber  used. 

7.  The  sill  of  a  barn  is  18'  long,  8"  wide,  and  10"  thick. 
Find  how  many  feet  of  lumber  the  sill  contains. 

Find  the  number  of  hoard  feet  in  each  of  the  following  lots : 

8.  12  boards,  each  16'  long,  10"  wide,  and  1"  thick. 

9.  42  planks,  each  14'  long,  1'  wide,  and  2"  thick. 

10.  46  boards,  each  16'  long,  8"  wide,  and  |"  thick. 
A  thickness  of  less  than  V  is  always  considered  as  V\ 

11.  68  joists,  each  14'  long,  8"  wide,  and  3"  thick. 


88  MENSURATION 

Exercises.    Areas  and  Volumes 

1 .  Find  the  area  of  the  elevation  of  the  angle  bracket  shown 
in  the  blueprint  on  page  89. 

The  term  **  elevation  "  is  used  extensively  in  connection  with  working 
drawings  and  building  plans.  Here  it  means  the  surface  shown  in  out- 
line in  the  blueprint,  that  is,  one  face  of  the  angle  bracket.  In  a  set  of 
plans  for  a  house  the  front  elevation  is  a  drawing,  not  in  perspective, 
showing  the  house  as  seen  from  the  front;  the  side  elevation  is  a 
similar  drawing  of  the  side;    and  so  on. 

2.  If  the  angle  bracket  is  made  of  cast  steel  such  that  a 
bar  ^'^  square  and  1'  long  weighs  2.5871b.,  how  much  does 
the  bracket  weigh? 

3.  In  planing  the  angle  bracket  a  machinist  removes  ^"  of 
metal  from  the  side  shown  in  the  drawing.  How  many  cubic 
inches  of  metal  does  he  remove? 

4.  The  cross  section  of  an  iron  joist  is  shown  in  the  blue- 
print.   Find  the  area  of  this  cross  section. 

The  rounding  of  the  corners  should  be  neglected  entirely. 

5.  In  Ex.  4  what  would  be  the  area  of  the  web  (the  part 
0.50''  thick)  if  the  thickness  were  0.875"  instead  of  0.50''? 

6.  How  many  square  inches  of  metal  would  be  removed  in 
planing  the  side  of  the  compound  rest? 

7.  The  slot  cleaner  shown  in  the  blueprint  is  made  of 
#16  B.  &  S.  gage  copper  weighing  2.302  lb.  per  square  foot 
of  surface.  Find  the  weight  of  the  cleaner.  At  38  (f  per 
pound,  what  is  the  value  of  the  metal  in  250  cleaners  ? 

8.  What  would  be  the  area  of  the  elevation  of  the  angle 
bracket  shown  in  the  blueprint  if  the  base  were  6  J",  the 
height  7f",  and  the  thickness  1|"? 

9.  In  Ex.  8  what  would  be  the  number  of  cubic  inches  in 
the  bracket  if  the  length  of  the  base  were  14"  ? 


AREAS  AND  VOLUMES 


89 


90  MENSUEATION 

Exercises.   Volumes 

1.  How  many  cubic  feet  of  lumber  are  there  in  15  planks, 
each  14^  6"  long,  6^'  wide,  and  2^'  thick  ?  How  many  board 
feet  are  there? 

2.  If  a  bar  of  iron  1"  square  and  1'  long  weighs  3.333  lb., 
what  is  the  weight  of  a  bar  2"  x  ^''  X  1'  ? 

3.  Using  the  data  given  in  Ex.  2,  find  the  weight  of  a 
bar  of  iron  if  the  dimensions  are  2^  x  |^'  X  3^ 

4.  If  cast  iron  weighs  0.260  lb.  per  cubic  inch,  what  is  the 
weight  of  a  bar  of  cast  iron  12'  4''  x  61-''  x  2"  ? 

5.  Find  the  number  of  cubic  inches  of  metal  in  the  angle 
plate  shown  on  page  91.  If  the  angle  plate  is  made  of  cast 
iron  weighing  450  lb.  per  cubic  foot,  what  is  the  cost  of  150 
plates  of  this  type  at  7J^  per  pound? 

6.  If  the  angle  plates  in  Ex.  5  are  finished  all  over  by 
machining  off  g\'',  that  is,  by  removing  ^^"  of  metal  from 
each  surface,  what  is  then  the  weight  of  150  plates  ? 

7.  The  step  block  is  made  of  steel  weighing  490  lb./cu.  ft. 
Find  the  weight  of  the  block. 

The  expression  "490  lb./cu.  ft."  means  "490  lb.  per  cubic  foot,"  and 
in  general  the  symbol  /  is  read  "  per  "  in  all  cases  of  this  kind. 

8.  At  $1.75  per  perch  of  24|  cu.  ft.,  find  the  cost  of  the 
stone  for  the  pier  shown  in  the  blueprint. 

The  perch  used  in  stone  work  is  generally  1  rd.  (16 1')  long,  1'  wide, 
and  11'  thick,  but  varies  locally,  being  often  taken  as  25  cu.  ft. 

9.  Allowing  575  bricks  per  cubic  yard,  and  5%  above  this 
for  waste  and  breakage,  how  many  bricks  would  be  required 
if  the  pier  in  Ex.  8  were  to  be  made  of  brick? 

10.  Taking  the  weight  of  copper  as  542  lb./cu.  ft.,  and  the 
cost  as  380  a  pound,  find  the  cost  of  125  stop  dogs  of 
the  type  shown  in  the  blueprint. 


VOLUMES 


91 


92 


MENSURATION 


Circumference  of  a  Circle.  If  we  measure  the  diameter  and 
the  cu'cumference  of  a  circle,  and  then  divide  the  circumference 
by  the  diameter,  we  shall  find  that 
circumference 


diameter 


=  3^,  approximately. 


The  number  3.1416  is  a  closer  approxiirmtion, 
and  3.14159  is  still  closer,  but  for  practical  work 
3},  -^2-,  or  3.14  is  used  unless  a  higher  degree  of 
accuracy  is  necessary. 

A  special  name  is  given  to  the  ratio  3{ ;  it  is  called  pi 
(written   tt,   a  Greek   letter).     That  is,   c:  d  =  7r,  or 

If  we  divide  these  two  equal  expressions  by  tt,  we  have 


Since  the  diameter  of  a  circle  is  twice  the  radius,  that  is, 

since  d  =  2r,  we  have 

c=2  7rr. 

If  we  divide  these  two  equal  expressions  by  2  it,  we  have 

_   c 

In  this  book  use  ^-,  of  3.14,  for  tt,  and  ^^,  or  0.32,  for  -,  in  all 
cases,  unless  otherwise  directed. 

Illustrative  Problems.    1.  Find  the  circumference  of  a  fly- 
wheel that  is  28''  in  diameter. 

c  =  Trd  =  -^j^-  X  28"  =  88". 

2.  What  radius  should  be  used  in  drawing  a  circle  for  the 
pattern  of  a  pulley  that  is  to  be  48.4"  in  circumference  ? 

1.1 


CIRCLES  98 

Exercises.   Circles 

1.  A  pulley  has  a  diameter  of  8".  Find  the  circumference 
to  the  nearest  0.1'^ 

2.  An  automobile  wheel  has  a  diameter  of  36".  Find  the 
radius  and  the  circumference. 

3.  A  locomotive  has  a  driving  wheel  6'  10"  in  diameter. 
Not  allowing  for  slipping,  how  far  will  the  locomotive  travel 
during  one  revolution  of  the  driving  wheel  ? 

4.  In  Ex.  3  how  many  R.P.M.  does  the  driving  wheel 
make  when  the  train  is  traveling  45  mi./hr.  ? 

5.  When  a  14-inch  steel  casting  is  being  turned  in  a  lathe, 
how  long  a  chip  is  cut  off  at  each  turn  of  the  casting? 

Consider  that  the  casting  is  a  cylinder  with  a  diameter  of  14"  and 
that  the  length  of  the  chip  is  equal  to  the  circumference. 

6.  Some  BX  cable  is  coiled  up  in  18  turns,  the  average 
diameter  of  the  coil  being  2'  4".    Find  the  length  of  the  cable. 

7.  The  flywheel  of  an  engine  is  14"  in  diameter  and  runs 
at  a  speed  of  150  R.P.M.  Find  the  rim  speed,  that  is,  the 
speed  of  a  point  on  the  rim,  in  F.P.M.  (feet  per  minute). 

8.  An  automobile  tire  is  34''  in  diameter.  At  what  speed 
should  the  wheel  revolve  in  order  that  the  car  may  have  a 
speed  of  18  mi./hr.? 

9.  A  boiler  is  6'  8"  in  diameter.  How  many  rivets  will 
there  be  in  a  circumferential  seam  if  the  rivet  holes  are 
spaced  approximately  3"  apart? 

10.  The  circumference  of  a  boiler  is  14'.  Find  the  diameter 
to  the  nearest  \", 

11.  In  making  a  drawing  of  a  wheel  that  is  to  have  a 
circumference  of  110",  the  scale  of  the  drawing  being  J^, 
what  length  of  radius  should  the  draftsman  use? 


94  MENSURATION 

12.  Find  the  circumference  of  the  pulley  shown  at  the  top 
of  the  blueprint  on  page  95. 

13.  If  the  diameter  of  the  pulley  were  half  as  large  as 
shown,  what  would  be  the  circumference  ? 

14.  If  the  circumference  of  the  pulley  were  1'  10",  what 
would  be  the  diameter? 

15.  If  a  pattern  maker  were  drawing  the  circle  for  a 
pulley  2'  9"  in  circumference,  what  radius  should  he  use  ? 

16.  The  circular  saw  shown  in  the  blueprint  has  35  teeth. 
Using  3.142  as  the  value  of  tt,  find  to  the  nearest  O.OOl"  the 
distance  between  the  points  of  adjacent  teeth. 

17.  If  in  Ex.  16  the  distance  between  the  points  of  adjacent 
teeth  were  1.571",  how  many  teeth  would  there  be?  Use 
3.1416  as  the  value  of  tt. 

18.  How  many  revolutions  will  the  automobile  wheel  make 
while  the  car  is  going  1.5  mi.?  3.9  mi.?  11.2  mi.? 

19.  In  Ex.  18  how  many  revolutions  would  there  be  in 
each  case  if  the  diameter  of  the  wheel  were  36"  ?  With 
which  size  of  wheel  would  there  be  the  less  wear  on  the 
tire  ?    Write  the  reason  for  your  answer. 

20.  Find  the  number  of  R.  P.  ^I.  made  by  a  wheel  34"  in 
diameter  on  a  car  that  is  going  at  the  rate  of  25  mi./hr. 

21.  On  a  certain  locomotive  the  drive  wheels  are  6'  in 
diameter  and  the  truck  wheels  are  2'  in  diameter.  Find  the 
number  of  R.P.M.  of  each  type  of  wheel  when  the  locomo- 
tive is  going  at  the  rate  of  40  mi./hr. 

22.  In  the  two  pulleys  connected  by  belting,  as  shown  in 
the  blueprint,  what  is  the  length  of  the  belting  ? 

23.  In  Ex.  22  what  would  be  the  length  of  the  belting 
if  the  diameter  of  each  pulley  were  twice  as  great? 


CIRCLES 


95 


96  MENSUKATION 

Area  of  a  Circle.  A  circle  may  be  separated  into  figures 
which  are  nearly  triangles,  the  height  of  each  triangle  being 
the  radius,  and  the  sum  of  the  bases  being  the  circumference. 
If  these  figures  were  exact  triangles,  the  area  of  the  circle 


would  be  ^  X  height  x  sum  of  bases ;  that  is,  the  area  would 
be  1^  X  radius  x  circumference.  It  is  proved  in  geometry  that 
this  is  the  true  area  of  a  circle. 

We  may  now  express  this  a.f  a  formula,  thus : 


A  =  lrc. 


Since  c  =  2  7rr,  A  =  ^^r  x  2  ttv, 


or  A  =  rrr^, 

d  .  .  * 

Since  -  =  r,  the  formula  A  =  irr^  may  be  written 

A-  — 

Illustrative  Problems.    1.  Find  the  area  of  a  circle  which 
is  drawn  with  a  radius  of  5". 

A  =  TTr^  =  3.14  X  5  X  5  =  78.5. 
Hence  the  area  of  the  circle  is  78.5  sq.  in. 

2.  Find  the  cross-section  area  of  a  shaft  14''  in  diameter. 

^        7 

^^^^22_x2£2^  =  154 

4  Jx^ 

Hence  the  cross-section  area  of  the  shaft  is  154  sq.  in. 


CIRCLES  97 

Exercises.   Circles 
Find  the  areas  of  circles^  given  the  radii  as  follows  : 
1.  r.  2.  14^  3.  1.4".  4.  4.9".  5.  56". 

6.  If  the  radius  of  one  circle  is  twice  as  long  as  the  radius 
of  another  circle,  how  do  the  circumferences  of  the  circles 
compare  ?    How  do  the  areas  compare  ? 

7.  If  one  circle  has  a  radius  three  times  as  long  as  the 
radius  of  another  circle,  how  do  the  circumferences  compare  ? 
How  do  the  areas  compare  ? 

8.  A  circular  mirror  is  2'  3"  in  diameter.  Find  the  cost 
of  resilvering  the  mirror  at  68<f  a  square  foot. 

9.  What  is  the  area  of  the  cross  section  of  a  water  pipe 
that  is  6"  in  diameter? 

10.  Find  the  entire  area  of  a  window  the  lower       /  \ 
part  of  which  is  a  rectangle  and  the  upper  part  a 
semicircle,  or  half  circle,  as  shown  in  the  figure. 

11.  In  a  park  there  is  a  circular  lake  240'  in 
diameter.    Find    the   number   of  square  yards   in 
a  walk  6'  wide  around  the  lake.   If  the  width  of  ^'^ 
the  walk  is  doubled,  what  is  then  the  area  of  the  walk  ? 

This  is  a  case  of  finding  the  area  of  a  ring.  If  we  let  R  be  the  radius 
of  the  outer  circle  and  7-  be  the  radius  of  the  inner  circle,  there  is  a 
convenient  formula  for  the  area,  which  is 

A=7r(R-\-r)(li-r). 

The  jiarentheses  indicate  that  we  must  first  add  R  and  r,  then  sub- 
tract r  from  R,  and  then  multiply  the  product  of  these  two  results  by  tt. 

12.  A  tree  the  cross  section  of  which  may  be  assumed  to 
be  a  circle  has  a  circumference  of  12'  3"  at  a  certain  height. 
What  is  the  area  of  the  top  of  the  stump  that  is  formed  by 
sawing  horizontally  through  the  tree  at  this  point  ? 


98  MENSURATION 

Measure  of  a  Cylinder.  We  frequently  need  to  find  the 
area  of  the  curve  surface  and  also  the  volume  of  a  cylinder. 

Tlie  area  of  the  curve  surface  of  a  cylinder  is  the  product 
of  TT  tiynes  the  diameter  and  the  height. 

Since  tt  times  the  diameter  is  tlie  circmiiference,  if  we 
think  of  the  curve  surface  of  the  cylinder  as  being  unrolled, 
we  see  that  the  product  of  the  circumference  and  the  height 
is  the  area  of  the  curve  surface. 

This  rule  is  too  long  for  practical  use,  and  so  we  write  it 

as  a  formula,  thus :  _         „ 

o  =  ndh. 

For  example,  suppose  that  we  wish  to  find  the  area  of  the 
curve  surface  of  a  pipe  15'  long  and  S"  in  diameter. 

Expressing  the  diameter  8"  as  0.25',  we  have 

^'  =  ird/i  =  :3.U  X  0.25  X  15  =  11.7750. 

Hence  the  area  of  the  curve  surface  is  approximately  11.78  sq.  ft. 

The  volume  of  a  cylinder  is  the  product  of  the  area  of  the  base 
and  the  height. 

We  write  this  rule  as  a  formula,  thus : 

V=bh. 

Since  the  area  of  the  base  is  7rr^  or  ^  7rd\  we  may  write 
the  formula  for  V  in  two  different  forms,  as  follows  : 

V=7rr^h, 
V=\TTd^h. 

4 

For  example,  suppose  that  we  wish  to  find  the  volume  of 
a  steel  shaft  3'  6''  long  and  ^"  in  diameter. 

Expressing  the  length  3'  6"  as  42",  we  have 

V  =  1  mlVi  =  ix  —  X;|x4x|;?  =  528. 
Hence  the  volume  of  the  shaft  is  528  cu.  in. 


CYLINDERS  99 

Exercises.    Cylinders 

1.  A  water  tank  is  20'  in  diameter'  and  16'  liigli,  inside 
measurements.    Find  the  area  of  the  interior  curve  surface. 

2.  Find  the  capacity  of  the  tank  in  Ex.  1. 

3.  A  piece  of  water  pipe  is  16'  long  and  the  internal 
diameter  is  4".  Find  the  number  of  cubic  inches  of  water 
required  to  fill  the  pipe. 

4.  A  cylindric  iron  pillar  supporting  a  ceiling  is  14' 
high  and  has  a  diameter  of  5".  If  the  pillar  is  solid  and 
weighs  441  lb./cu.  ft.,  what  is  its  weight  ? 

5.  The  cross  section  of  a  hollow  cylindric  iron  pillar  is  as 
here  shown.     The  external  diameter  is   6",   the 
internal  diameter  4",  and  the  length  12'.    If  the 
grade  of  iron   used  weighs  441  lb./cu.  ft.,   what 
is  the  weight  of  the  pillar  ? 

The  formula  for  the  area  of  a  ring  was  given  on  page  97.  From 
this  formula  it  is  evident  that  if  we  are  dealing  with  a  hollow  cylinder 
the  formula  for  the  volume  is 

V=7r(R  +  r)(R-r)h, 

where  R  is  the  external  radius,  r  the  internal  radius,  and  h  the  height, 
or  length,  of  the  cylinder. 

6.  In  finding  the  strength  of  an  iron  rod  the  cross-section 
area  is  required.    Find  this  area  for  a  rod  3i"  in  diameter. 

7.  A  tinsmith  cuts  out  100  circular  pieces  of  tin  for  the 
bottoms  of  cups,  using  a  radius  of  2".  How  many  square 
inches  of  tin  are  there  in  all  the  pieces  ? 

8.  The  number  of  pounds  which  can  be  supported  at  the 
center  of  the  distance  between  the  supports  of  a  cylindric  cast- 
iron  shaft  /  feet  long  and  d  inches  in  diameter  is  500  dyi.  Find 
the  weight  which  can  be  thus  supported  by  a  shaft  which  is  3" 
in  diameter  and  has  a  length  of  16'  between  the  supports. 


100  MENSURATION 

9.  The  blind  collar  shown  on  page  101  is  made  of  cast 
iron  weighing  450  Ib./cu.  ft.  Find  the  weight  of  the  collar, 
the  hole  running  only  part  of  the  length,  as  shown. 

10.  In  Ex.  9  what  would  be  the  weight  of  the  collar  if  the 
hole  ran  all  the  way  through  it  ? 

11.  Before  the  bronze  bushing  was  turned  to  its  present 
dimensions  there  was  |'^  left  on  the  entire  outside  surface, 
including  the  ends,  for  finishing.  An  automobile  factory  uses 
2500  of  these  bushings  per  day.  If  the  scrap  is  worth  35 (f 
per  pound  and  bronze  weighs  529  Ib./cu.  ft.,  how  much  is  the 
value  of  the  scrap  for  a  day  ? 

12.  The  jig  bushing  shown  in  the  blueprint  is  turned  and 
drilled  from  li|-inch  round  stock.  How  many  pounds  of 
steel  weighing  490  Ib./cu.  ft.  are  required  in  manufacturing 
75  bushings  of  this  type? 

13.  Find  the  weight  of  the  steel  which  is  wasted  in  turning 
and  drilling  the  lot  mentioned  in  Ex.  12. 

14.  The  pattern  for  the  bearing  cap  is  made  of  mahogany 
weighing  53  Ib./cu.  ft.  If  the  bearing  cap  is  made  of  cast 
iron  weighing  450  Ib./cu.  ft.,  the  cap  is  how  many  pounds 
heavier  than  the  pattern? 

The  slight  shrinkage  in  casting  may  be  neglected. 

15.  A  factory  orders  125  cast-iron  shaft  supports  and  25 
bronze  supports  of  the  pattern  shown  in  the  blueprint.  The 
cost  of  the  cast-iron  supports  is  8^  a  pound,  and  that  of 
the  bronze  supports  is  38(f  a  pound.  Taking  the  weights  of 
the  metals  as  given  in  Exs.  9  and  11,  find  the  entire  cost. 

16.  How  many  cubic  inches  of  metal  are  there  in  the  bronze 
bushing  shown  in  the  blueprint  ?  How  many  would  there  be 
if  all  the  dimensions  were  doubled  ?  if  they  were  multiplied 
by  three  ?  if  they  were  half  the  size  given  ? 


CYLINDERS 


101 


102 


MENSUKATION 


Meaning  of  Roots.  In  engineering  and  in  various  lines  of 
industry  it  is  often  necessary  to  use  the  roots  of  numbers, 
particularly  the  square  root. 

If  a  number  is  the  product  of  two  equal  factors,  either 
factor  is  called  the  square  root  of  the  number.  For  example, 
because  25  =  5  x  5,  we  say  that  5  is  the  square  root  of  25,  and 
we  indicate  this  by  the  symbol  V     .    Thus  5  =  V25. 

Accordmg  to  this  definition  3  would  have  no  square  root, 
because  3  has  not  two  equal  factors.  We  therefore  extend 
the  idea  of  root  to  include  approximate  factors,  and  say  that 
V3  =  1.732  to  four  significant  figures  or  to  three  decimal 
places,  because  1.7322=3,  approximately. 

Approximate  roots  being  allowed  as  in  the  case  of  the  square  root, 
if  a  number  is  the  product  of  three  equal  factors,  any  one  of  these 
factors  is  called  the  cube  root  of  the  number ;  if  of  four  equal  factors,  the 
fourth  root-,  and  so  on.  Thus  the  cube  root  of  8,  written  v  8,  is  2,  the 
index  of  the  root  being  3 ;  and  ■\^  is  3,  the  index  of  the  root  being  4. 

In  practice  all  roots  are  found  by  means  of  tables,  and 
hence  we  shall  give  but  little  attention  to  the  older  methods. 
We  shall  pay  some  attention  to  finding  square  roots  because 
they  are  the  ones  that  the  student  will  most  often  need. 

Square  of  a  Sum.  If  we  wish  to  find  the  square  of  25 
we  may,  if  desired,  multiply  20  -f-  5  by  itself,  as  here  shown. 

Thus,  if  we 
take  20  as  the 
first  part  and 
5  as  the  sec- 
ond part  of 
25,  the  square 
of  25  consists 
of  the  follow- 
ing three  parts,  as  shown  above:  (1)  the  square  of  20, 
(2)  twice  the  product  of  20  and  5,  and  (3)  the  square  of  5. 


20  +  5 

20  +  5 

Product  by  5 

20  X  5  4-  52 

Product  by  20 

202+        20x5 

Total  product 

202  H-  2  X  20  X  5  +  52 

SQUARE  ROOT  103 

Square  Root.  In  tlie  preceding  work,  if  we  put  t  for  the 
tens  and  u  for  the  units,  we  see  that 

The  square  of  a  iiumher  coyitains  the  square  of  the  teus^plus  twice 
the  product  of  the  tens  and  units,  plus  the  square  of  the  units. 

Since  1  =  12,  100=102,  10,000  =  1002,  the  square  root  of 
any  number  between  1  and  100  lies  between  1  and  10,  and 
the  square  root  of  any  number  between  100  and  10,000  lies 
between  10  and  100.  In  other  words,  the  integral  part  of 
the  square  root  of  any  whole  number  of  one  or  two  figures  is 
a  number  of  one  figure ;  that  of  any  w^hole  number  of  three 
or  four  figures  is  a  number  of  two  figures ;  and  so  on. 

If,  therefore,  a  whole  number  is  separated  into  periods  of 
two  figures  each,  the  number  of  figures  in  the  integral  part  of 
the  square  root  is  equal  to  the  number  of  periods.  Since  we 
mark  off  the  periods  from  riglit  to  left,  the  period  at  tlie  left 
may  have  one  or  two  figures ;  for  example,  22  09  and  7  89  04. 

Illustrative  Problem.    Find  the  square  root  of  3481. 

Separate  the  figures  of  the  number  into  periods  of  two  figures  each, 
beginning  at  the  right. 

The  first  period  contains  the  square  of  the  tens'  number  of  the  root. 

Since  the  greatest  square  in  34  is  25,  then  5,  the     

square  root  of  25,  is  the  tens'  figure  of  the  root. 

Subtracting  the  square  of  the  tens,  the  re- 
mainder contains  twice  the  tens  times  the  units, 
plus  the  square  of  the  units.  Dividing  by  twice 
the  tens  (that  is,  by  100,  which  is  2  x  5  tens),  we 
find,  approximately,  the  units'  figure.  Dividing 
981  by  100,  we  have  9  as  the  units'  figure. 

Since  twice  the  tens  times  the  units,  plus  the 
square  of  the  units,  is  equal  to  (twice  the  tens  plus  the  units)  times  the 
units,  that  is,  since  2  x  50  x  9  +  9^  =  (2  x  50  +  9)  x  9,  we  add  9  to  100  and 
multiply  the  sum  by  9.    The  product  is  981,  and  there  is  no  remainder. 
Hence  the  square  root  is  59.    Checking  the  work,  59-  =  o48l. 


34  81(59 
25 
100 


109 


9  81 

9  81 


104 


MENSURATION 


Square  Root  with  Decimals.  In  finding  the  square  root  of 
a  number  wliich  contains  a  decimal  point  we  proceed  in  the 
same  way  as  before,  except  that  we  separate  the  number  into 
periods  of  two  figures  each,  beginning  at  the  decimal  point. 

For  example,  find  the  square  root  of  151.29. 

Separate  the  figures  into  periods,  beginning  at  the  decimal  point. 

The  greatest  square  of  the  tens  in  151.29  is  100,  and  the  square 
root  of  100  is  10. 

Then  51.29  contains  2  xlO  times  the 
units'  number  of  the  root,  plus  the  square 
of  the  units'  number. 

Dividing  51  by  2  x  10,  or  20,  we  find 
that  the  next  figure  of  the  root  is  2. 

We  have  now  found  12,  the  square  being 
100  +  40  +  4,  or  144. 

Then  7.29  contains  2  x  12  times  the 
tenths'  number  of  the  root,  plus  the  square 
of  the  tenths'  number,  because  we  have 
subtracted  144,  which  is  the  square  of  12. 

Dividing  by  240,  we  find  that  the  tenths'  figure  of  the  root  is  3. 

There  is  no  remainder,  and  hence  the  square  root  of  151.29  is  12.3. 

If  the  number  is  not  a  perfect  square  we  may  annex  pairs  of  zeros 
at  the  right  and  find  the  root  to  as  many  decimal  places  as  we  choose. 


161.29(12.3 

1 

20 
22 

61 
44 

240 
243 

7  29 
7  29 

Exercises.    Square  Root 
Find  the  square  root  of  each  of  the  following 


625. 

6.25. 

0.0625. 

324. 

3.24. 

0.0324. 

484. 


8.  0.0484, 


9.  729. 

10.  7.29. 

11.  0.0729. 

12.  2025. 

13.  20.25. 

14.  0.2025. 

15.  0.1296. 

16.  0.1089. 


17.  1225. 

18.  12.25. 

19.  0.1225. 

20.  4096. 

21.  40.96. 

22.  4.096. 

23.  14,641. 

24.  146.41. 


SQUARE  EOOT 


105 


Square  on  the  Hypotenuse.    In  a  right  triangle  the  side  op- 
posite the  right  angle  is  called  the  hypotenuse  of  the  triangle. 

If  a  floor  is  paved  with  triangular  tiles  as  in  this  figure,  it  is 
easy  to  mark  out  a  right  triangle,  as  shown  by  the  heavy  lines. 
It  is  seen  that  the  square  on  the  hypotenuse  contains  eight 
small  triangles,  while  each  square  on  a 
side  contains  four  such  triangles.   Hence 

The  square  on  the  hypotenuse  is  the  sum 
of  the  squares  on  the  other  two  sides. 

Letting  the  hypotenuse  be  h  and  the 
other  sides  be  a  and  ?>,  we  have  the  fol- 
lowing formulas : 

Illustrative  Problems.    1.  Given  that  a  =  9"  and  I>  =  \2"  in 
a  right  triangle,  find  the  value  of  h. 

h 


Va2  +  ^2  =  V92  +  I22  =  V225  =  15. 
Hence  the  length  of  the  hypotenuse  is  15''. 

2.  In  a  right  triangle  it  is  given  that  A=  20^  and  h=12'. 
Find  the  value  of  a. 


a  =  VA2  _  /,2  =  V202  -  122=  V256 
Hence  the  length  of  side  a  is  16'. 


16. 


Exercises.    Square  Root 
Find  the  sides  of  squares  that  have  the  following  areas  : 

1.  3136  sq.ft.  3.  6561  sq.ft.  5.  8281  sq. yd. 

2.  3844  sq.ft.  4.  6889  sq.  ft.  6.  9409  sq.  yd. 

7.  If  the  hypotenuse  of  a  right  triangle  is  60"  and  one 
side  is  36'',  what  is  the  length  of  the  other  side  ? 


106 


MENSURATION 


8.  How  long  is  the  diagonal  of  a  floor  48'  by  64'? 

In  those  examples  on  this  page  which  involve  numbers  that  are  not 
perfect  squares  find  each  square  root  to  the  nearest  hundredth. 

9.  Find  the  length  of  the  diagonal  of  a  square  that  eon- 
tains  13  sq.  ft. 

10.  P^ind  the  length,  between  the  points  where  it  is  attached, 
of  a  wire  drawn  taut  from  the  top  of  a  70-foot  flagpole  to 
a  stake  set  in  the  ground  30'  from  the  foot  of  the  pole. 

11.  A  wire  is  to  be  fastened  to  a  telegraph  pole  at  a 
point  20'  above  the  ground  and  is  to  be  stretched  taut  to  a 
stake  in  the  ground  15'  6"  from  the  foot,  so  as  to  hold  the 
pole  perpendicular.  If  5'  is  allowed  for  fastening  the  wire  at 
each  end,  find  the  length  of  wu^e  required. 

12.  The  arm  of  a  derrick  for  hoisting  coal 
is  27'  6"  long  and  swings  over  an  opening 
22'  from  the  base  of  the  derrick.  How  far 
is  the  top  of  the  arm  above  the  opening? 

13.  The  foot  of  a  48-foot  ladder  is  16'  from  the  wall  o; 
a  building  against  Avhich  the  top  rests.  How  high  does  the 
ladder  reach  on  the  wall  ? 

14.  To  find  the  length  of  this  pond  a 
group  of  students  laid  off  the  right  tri- 
angle ACB^  as  shown.  They  found  by 
measuring  that  ^C=428',  i?C=321',  and 
AD  =76'.    Calculate  the  length  of  BB. 

15.  How  far  from  the  wall  of  a  house  must,  the  foot  o 
a  36-foot  ladder  be  placed  so  that  the  top  of  the  ladder  maj 
touch  a  window  sill  32'  above  the  ground  ? 

16.  A  square  building  lot  has  an  area  of  17,500  sq.  ft 
P^ind  the  perimeter  of  the  lot  to  the  nearest  0.1'.  Find  th< 
perimeter  of  a  lot  with  four  times  this  area. 


SQUARE  liOOT  107 

Table  of  Square  Roots.  People  can  no  longer  afford  the 
time  to  find  square  roots  by  the  method  on  pages  103  and 
104.    They  use  such  a  table  as  the  one  on  pages  108-111. 

In  that  table  the  first  two  figures  of  the  number  whose 
root  we  seek  are  given  at  the  left  in  the  column  marked  N, 
and  the  third  figure  is  given  at  the  top.  For  example,  on 
page  108  the  square  root  of  1.00  is  1.000,  this  being  opposite 
the  number  1.0  and  under  0 ;  the  square  root  of  1.01  is 
1.005,  this  being  opposite  the  number  1.0  and  under  1 ;  the 
square  root  of  1.02  is  1.010  ;  and  so  on. 

Furthermore,  since  VlOO=10,  multiplying  a  number  by 
100  multiplies  the  square  root  by  10.    Thus: 


Vl.76  =  1.327.  V8.38  =  2.895. 

VT76  =13.27.  V838  =  28.95. 


V17600  =  132.7.  V83800  =  289.5. 

On  each  page  of  the  table  the  right-hand  columns  headed 
1,  2,  3,  ...,  9  show  the  numbers  to  be  added  to  the  root 
when  the  number  whose  square  root  is  desired  contains 
a  fourth  figure.  For  example,  on  page  108  we  find  that 
V2^  =1.637;  but  if  we  wish  to  find  V2.687  we  look  along 
the  line  from  2.6  and  find  2  in  the  right-hand  column  under  7. 
This  means  that  we  should  add  0.002  to  1.637,  giving 
V2.687  =  1.639.    Consider  also  the  following: 

V934.9  =  30.56  +  0.01  =  30.57. 
V81.55  =  9.028  +  0.003  =  9.031. 
V8155  =  90.28  +  0.03  =  90.31. 

The  last  two  roots  above  are  found  on  page  111. 

Thus,  the  table  gives  the  square  roots  of  all  numbers  from 
1  to  9999.  The  table  on  page  112  gives  on  one  page,  for 
easy  reference,  the  squares,  cubes,  square  roots,  and  cube 
roots  of  all  whole  numbers  from  1  to  100. 


108 


MENSURATION 


SQUARE  KOOTS  OF  NUMBERS  FROM  1  TO  9.999 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

123 

456 

789 

1.0 

1.000 

1.005 

1.010 

1.015 

1.020 

1.025 

1.030 

1.034 

1.039 

1.044 

Oil 

223 

344 

1.1 

1.049 

1.054 

1.058 

1.063 

1.068 

1.072 

1.077 

1.082 

1.086 

1.091 

Oil 

223 

344 

1.2 

1.095 

1.100 

1.105 

1.109 

1.114 

1.118 

1.122 

1.127 

1.131 

1.136 

Oil 

223 

344 

1.3 

1.140 

1.145 

1.149 

1.153 

1.158 

1.162 

1.166 

1.170 

1.175 

1.179 

oil 

223 

334 

1.4 

1.183 

1.187 

1.192 

1.196 

1.200 

1.204 

1.208 

1.212 

1.217 

1.221 

Oil 

222 

334 

1.5 

1.225 

1.229 

1.233 

1.237 

1.241 

1.245 

1.249 

1.253 

1.257 

1.261 

oil 

222 

334 

1.6 

1.265 

1.269 

1.273 

1.277 

1.281 

1.285 

1.288 

1.292 

1.296 

1.300 

oil 

222 

334 

1.7 

1.304 

1.308 

1.311 

1.315 

1.319 

1.323 

1.327 

1.330 

1.334 

1.338 

oil 

222 

333 

1.8 

1.342 

1.345 

1.349 

1.353 

1.356 

1.360 

1.364 

1.367 

1.371 

1.375 

oil 

122 

333 

1.9 

1.378 

1.382 

1.386 

1.389 

1.393 

1.396 

1.400 

1.404 

1.407 

1.411 

oil 

122 

333 

2.0 

1.414 

1.418 

1.421 

1.425 

1.428 

1.432 

1.435 

1.439 

1.442 

1.446 

oil 

122 

233 

2.1 

1.449 

1.453 

1.456 

1.459 

1.463 

1.466 

1.470 

1.473 

1.476 

1.480 

oil 

122 

233 

2.2 

1.483 

1.487 

1.490 

1.493 

1.497 

1.500 

1.503 

1.507 

1.510 

1.513 

oil 

122 

233 

2.3 

1.517 

1.520 

1.523 

1.526 

1.530 

1.533 

1.536 

1.539 

1.543 

1.546 

oil 

122 

233 

2.4 

1.549 

1.552 

1.556 

1.559 

1.562 

1.565 

1.568 

1.572 

1.575 

1.578 

oil 

122 

233 

2.5 

1.581 

1.584 

1.587 

1.591 

1.594 

1.597 

1.600 

1.603 

1.606 

1.609 

oil 

122 

233 

2.6 

1.612 

1.616 

1.619 

1.622 

1.625 

1.628 

1.631 

1.634 

1.637 

1.640 

oil 

122 

223 

2.7 

1.643 

1.646 

1.649 

1.652 

1.655 

1.658 

1.661 

1.664 

1.667 

1.670 

oil 

122 

223 

2.8 

1.673 

1.676 

1.679 

1.682 

1.685 

1.688 

1.691 

1.694 

1.697 

1.700 

oil 

112 

223 

2.9 

1.703 

1.706 

1.709 

1.712 

1.715 

1.718 

1.720 

1.723 

1.726 

1.729 

oil 

112 

223 

3.0 

1.732 

1.735 

1.738 

1.741 

1.744 

1.746 

1.749 

1.752 

1.755 

1.758 

oil 

112 

223 

3.1 

1.761 

1.764 

1.766 

1.769 

1.772 

1.775 

1.778 

1.780 

1.783 

1.786 

oil 

112 

223 

3.2 

1.789 

1.792 

1.794 

1.797 

1.800 

1.803 

1.806 

1.808 

1.811 

1.814 

oil 

112 

2  22 

3.3 

1.817 

1.819 

1.822 

1.825 

1.828 

1.830 

1.833 

1.836 

1.838 

1.841 

oil 

112 

222 

3.4 

1.844 

1.847 

1.849 

1.852 

1.855 

1.857 

1.860 

1.863 

1.865 

1.868 

oil 

112 

222 

3.5 

1.871 

1.873 

1.876 

1.879 

1.881 

1.884 

1.887 

1.889 

1.892 

1.895 

oil 

112 

222 

3.6 

1.897 

1.900 

1.903 

1.905 

1.908 

1.910 

1.913 

1.916 

1.918 

1.921 

oil 

112 

222 

3.7 

1.924 

1.926 

1.929 

1.931 

1.934 

1.936 

1.939 

1.942 

1.944 

1.947 

oil 

112 

222 

3.8 

1.949 

1.952 

1.954 

1.957 

1.960 

1.962 

1.965 

1.967 

1.970 

1.972 

oil 

112 

222 

3.9 

1.975 

1.977 

1.980 

1.982 

1.985 

1.987 

1.990 

1.992 

1.995 

1.997 

oil 

112 

222 

4.0 

2.000 

2.002 

2.005 

2.007 

2.010 

2.012 

2.015 

2.017 

2.020 

2.022 

001 

111 

222 

4.1 

2.025 

2.027 

2.030 

2.032 

2.035 

2.037 

2.040 

2.042 

2.045 

2.047 

001 

111 

222 

4.2 

2.049 

2.052 

2.054 

2.057 

2.059 

2.062 

2.064 

2.066 

2.069 

2.071 

001 

111 

222 

4.3 

2.074 

2.076 

2.078 

2.081 

2.083 

2.086 

2.088 

2.090 

2.093 

2.095 

001 

111 

222 

4.4 

2.098 

2.100 

2.102 

2.105 

2.107 

2.110 

2.112 

2.114 

2.117 

2.119 

001 

111 

222 

4.5 

2.121 

2.124 

2.126 

2.128 

2.131 

2.133 

2.135 

2.138 

2.140 

2.142 

001 

111 

222 

4.6 

2.145 

2.147 

2.149 

2.152 

2.154 

2.156 

2.159 

2.161 

2.163 

2.166 

001 

111 

222 

4.7 

2.168 

2.170 

2.173 

2.175 

2.177 

2.179 

2.182 

2.184 

2.186 

2.189 

001 

111 

222 

4.8 

2.191 

2.193 

2.195 

2.198 

2.200 

2.202 

2.205 

2.207 

2.209 

2.211 

001 

111 

222 

4.9 

2.214 

2.216 

2.218 

2.220 

2.223 

2.225 

2.227 

2.229 

2.232 

2.234 

001 

111 

222 

50 

2.236 

2.238 

2.241 

2.243 

2.245 

2.247 

2.249 

2.252 

2.254 

2.256 

001 

111 

222 

5.1 

2.258 

2.261 

2.263 

2.265 

2.267 

2.269 

2.272 

2.274 

2.276 

2.278 

001 

111 

222 

5.2 

2.280 

2.283 

2.285 

2.287 

2.289 

2.291 

2.293 

2.296 

2.298 

2.300 

001 

111 

222 

5.3 

2.302 

2.304 

2.307 

2.309 

2.311 

2.313 

2.315 

2.317 

2.319 

2.322 

001 

111 

222 

5.4 

2.324 

2.326 

2.328  2.330 

2.332 

2.335 

2.337 

2.339 

2.341 

2.343 

001 

111 

122 

TABLE  OF  SQUARE  ROOTS 


109 


SQUARE  ROOTS  OF  NUMBERS  FROM  1  TO  9.999 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

6.5 

2.345 

2.347 

2.349 

2.352 

2.354 

2.356 

2.358 

2.360 

2.362 

2.364 

001 

122 

5.6 

2.366 

2.369 

2.371 

2.373 

2.375 

2.377 

2.379 

2.381 

2.383 

2.385 

001 

122 

5.7 

2.387 

2.390 

2.392 

2.394 

2.396 

2.398 

2.400 

2.402 

2.404 

2.406 

001 

122 

5.8 

2.408 

2.410 

2.412 

2.415 

2.417 

2.419 

2.421 

2.423 

2.425 

2.427 

001 

122 

5.9 

2.429 

2.431 

2.433 

2.435 

2.437 

2.439 

2.441 

2.443 

2.445 

2.447 

001 

122 

6.0 

2.449 

2.452 

2.454 

2.056 

2.458 

2.460 

2.462 

2.464 

2.466 

2.468 

001 

122 

6.1 

2.470 

2.472 

2.474 

2.476 

2.478 

2.480 

2.482 

2.484 

2.486 

2.488 

001 

122 

6.2 

2.490 

2.492 

2.494 

2.496 

2.498 

2.500 

2.502 

2.504 

2.506 

2.508 

001 

122 

6.3 

2.510 

2.512 

2.514 

2.516 

2.518 

2.520 

2.522 

2.524 

2.526 

2.528 

001 

122 

6.4 

2.530 

2.532 

2.534 

2.536 

2.538 

2.540 

2.542 

2.544 

2.546 

2.548 

001 

122 

6.6 

2.550 

2.551 

2.553 

2.555 

2.557 

2.559 

2.561 

2.563 

2.565 

2.567 

001 

122 

6.6 

2.569 

2.571 

2.573 

2.575 

2.577 

2.579 

2.581 

2.583 

2.585 

2.587 

001 

122 

6.7 

2.588 

2.590 

2.592 

2.594 

2.596 

2.598 

2.600 

2.602 

2.604 

2.606 

001 

122 

6.8 

2.608 

2.610 

2.612 

2.613 

2.615 

2.617 

2.619 

2.621 

2.623 

2.625 

001 

122 

6.9 

2.627 

2.629 

2.631 

2.632 

2.634 

2.636 

2.638 

2.640 

2.642 

2.644 

001 

122 

7.0 

2.646 

2.648 

2.650 

2.651 

2.653 

2.655 

2.657 

2.659 

2.661 

2.663 

001 

122 

7.1 

2.665 

2.666 

2.668 

2.670 

2.672 

2.674 

2.676 

2.678 

2.680 

2.681 

001 

112 

7.2 

2.683 

2.685 

2.687 

2.689 

2.691 

2.693 

2.694 

2.696 

2.698 

2.700 

001 

112 

7.3 

2.702 

2.704 

2.706 

2.707 

2.709 

2.711 

2.713 

2.715 

2.717 

2.718 

001 

112 

7.4 

2.720 

2.722 

2.724 

2.726 

2.728 

2.729 

2.731 

2.733 

2.735 

2.737 

001 

112 

7.5 

2.739 

2.740 

2.742 

2.744 

2.746 

2.748 

2.750 

2.751 

2.753 

2.755 

001 

112 

7.6 

2.757 

2.759 

2.760 

2.762 

2.764 

2.766 

2.768 

2.769 

2.771 

2.773 

001 

112 

7.7 

2.775 

2.777 

2.778 

2.780 

2.782 

2.784 

2.786 

2.787 

2.789 

2.791 

001 

112 

7.8 

2.793 

2.795 

2.796 

2.798 

2.800 

2.802 

2.804 

2.805 

2.807 

2.809 

001 

112 

7.9 

2.811 

2.812 

2.814 

2.816 

2.818 

2.820 

2.821 

2.823 

2.825 

2.827 

001 

112 

8.0 

2.828 

2.830 

2.832 

2.834 

2.835 

2.837 

2.839 

2.841 

2.843 

2.844 

001 

112 

8.1 

2.846 

2.848 

2.850 

2.851 

2.853 

2.855 

2.857 

2.858 

2.860 

2.862 

001 

112 

8.2 

2.864 

2.865 

2.867 

2.869 

2.871 

2.872 

2.874 

2.876 

2.877 

2.879 

001 

112 

8.3 

2.881 

2.883 

2.884 

2.886 

2.888 

2.890 

2.891 

2.893 

2.895 

2.897 

001 

112 

8.4 

2.898 

2.900 

2.902 

2.903 

2.905 

2.907 

2.909 

2.910 

2.912 

2.914 

001 

112 

8.6 

2.915 

2.917 

2.919 

2.921 

2.922 

2.924 

2.926 

2.927 

2.929 

2.931 

001 

112 

8.6 

2.933 

2.934 

2.936 

2.938 

2.939 

2.941 

2.943 

2.944 

2.946 

2.948 

001 

112 

8.7 

2.950 

2.951 

2.953 

2.955 

2.956 

2.958 

2.960 

2.961 

2.963 

2.965 

001 

112 

8.8 

2.966 

2.968 

2.970 

2.972 

2.973 

2.975 

2.977 

2.978 

2.980 

2.982 

001 

112 

8.9 

2.983 

2.985 

2.987 

2.988 

2.990 

2.992 

2.993 

2.995 

2.997 

2.998 

001 

112 

9.0 

3.000 

3.002 

3.003 

3.005 

3.007 

3.008 

3.010 

3.012 

3.013 

3.015 

000 

111 

9.1 

3.017 

3.018 

3.020 

3.022 

3.023 

3.025 

3.027 

3.028 

3.030 

3.032 

000 

111 

9.2 

3.033 

3.035 

3.036 

3.038 

3.040 

3.041 

3.043 

3.045 

3.046 

3.048 

000 

111 

9.3 

3.050 

3.051 

3.053 

3.055 

3.056 

3.058 

3.059 

3.061 

3.063 

3.064 

000 

111 

9.4 

3.066 

3.068 

3.069 

3.071 

3.072 

3.074 

3.076 

3.077 

3.079 

3.081 

000 

111 

9.5 

3.082 

3.084 

3.085 

3.087 

3.089 

3.090 

3.092 

3.094 

3.095 

3.097 

000 

111 

9.6 

3.098 

3.100 

3.102 

3.103 

3.105 

3.106 

3.108 

3.110 

3.111 

3.113 

000 

ill 

9.7 

3.114 

3.116 

3.118 

3.119 

3.121 

3.122 

3.124 

3.126 

3.127 

3.129 

000 

111 

9.8 

3.130 

3.132 

3.134 

3.135 

3.137 

3.138 

3.140 

3.142 

3.143 

3.145 

000 

111 

9.9 

3.146 

3.148 

3.150 

3.151 

3.153 

3.154 

3.156 

3.158 

3.159 

3.161 

000 

111 

no 


MENSURATION 


SQUARE  ROOTS  OF  NUMBERS  FROM  10  TO  99.99 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

7 

^ 

10 

3.162 

3.178 

3.194 

3.209 

3.225 

3.240 

3.256 

3.271 

3.286 

3.302 

235 

689 

11  12  14 

11 

3.317 

3.332 

3.347 

3.362 

3.376 

3.391 

3.406 

3.421 

3.435 

3.450 

134 

679 

10  12  13 

12 

3.464 

3.479 

3.493 

3.507 

3.521 

3.536 

3.550 

3.564 

3.578 

3.592 

134 

678 

10  11  13 

13 

3.606 

3.619 

3.633 

3.647 

3.661 

3.674 

3.688 

3.701 

3.715 

3.728 

134 

578 

10  11  12 

14 

3.742 

3.755 

3.768 

3.782 

3.795 

3.808 

3.821 

3.834 

3.847 

3.860 

134 

578 

9  1112 

15 

3.873 

3.886 

3.899 

3.912 

3.924 

3.937 

3.950 

3.962 

3.975 

3.987 

134 

568 

91011 

16 

4.000 

4.012 

4.025 

4.037 

4.050 

4.062 

4.074 

4.087 

4.099 

4.111 

124 

567 

9  10  11 

17 

4.123 

4.135 

4.147 

4.159 

4.171 

4.183 

4.195 

4.207 

4.219 

4.231 

124 

567 

8  1011 

18 

4.243 

4.254 

4.266 

4.278 

4.290 

4.301 

4.313 

4.324 

4.336 

4.347 

123 

567 

8 

9  10 

19 

4.359 

4.370 

4.382 

4.393 

4.405 

4.416 

4.427 

4.438 

4.450 

4.461 

123 

567 

8 

9  10 

20 

4.472 

4.483 

4.494 

4.506 

4.517 

4.528 

4.539 

4.550 

4.561 

4.572 

123 

467 

8 

910 

21 

4.583 

4.593 

4.604 

4.615 

4.626 

4.637 

4.648 

4.658 

4.669 

4.680 

123 

456 

8 

9  10 

22 

4.690 

4.701 

4.712 

4.722 

4.733 

4.743 

4.754 

4.764 

4.775 

4.785 

123 

456 

7 

8    9 

23 

4.796 

4.806 

4.817 

4.827 

4.837 

4.848 

4.858 

4.868 

4.879 

4.889 

123 

456 

7 

8    9 

24 

4.899 

4.909 

4.919 

4.930 

4.940 

4.950 

4.960 

4.970 

4.980 

4.990 

123 

456 

7 

8    9 

25 

5.000 

5.010 

5.020 

5.030 

5.040 

5.050 

5.060 

5.070 

5.079 

5.089 

123 

456 

7 

8    9 

26 

5.099 

5.109 

5.119 

5.128 

5.138 

5.148 

5.158 

5.167 

5.177 

5.187 

123 

456 

7 

8    9 

27 

5.196 

5.206 

5.215 

5.225 

5.235 

5.244 

5.254 

5.263 

5.273 

5.282 

123 

416 
4?6 

7 

8    9 

28 

5.292 

5.301 

5.310 

5.320 

5.329 

5.339 

5.348 

5.357 

5.367 

5.376 

123 

7 

7    8 

29 

5.385 

5.394 

5.404 

5.413 

5.422 

5.431 

5.441 

5.450 

5.459 

5.468 

123 

455 

6 

7    8 

30 

5.477 

5.486 

5.495 

5.505 

5.514 

5.523 

5.532 

5.541 

5.550 

5.559 

123 

445 

6 

7    8 

31 

5.568 

5.577 

5.586 

5.595 

5.604 

5.612 

5.621 

5.630 

5.639 

5.648 

123 

345 

6 

7    8 

32 

5.657 

5.666 

5.675 

5.683 

5.692 

5.701 

5.710 

5.718 

5.727 

5.736 

123 

345 

6 

7    8 

33 

5.745 

5.753 

5.762 

5.771 

5.779 

5.788 

5.797 

5.805 

5.814 

5.822 

123 

345 

6 

7    8 

34 

5.831 

5.840 

5.848 

5.857 

5.865 

5.874 

5.882 

5.891 

5.899 

5.908 

123 

345 

6 

7    8 

35 

5.916 

5.925 

5.933 

5.941 

5.950 

5.958 

5.967 

5.975 

5.983 

5.992 

122 

345 

6 

7    8 

36 

6.000 

6.008 

6.017 

6.025 

6.033 

6.042 

6.050 

6.058 

6.066 

6.075 

122 

345 

6 

7    7 

37 

6.083 

6.091 

6.099 

6.107 

6.116 

6.124 

6.132 

6.140 

6.148 

6.156 

122 

345 

6 

7    7 

38 

6.164 

6.173 

6.181 

6.189 

6.197 

6.205 

6.213 

6.221 

6.229 

6.237 

122 

345 

6 

6    7 

39 

6.245 

6.253 

6.261 

6.269 

6.277 

6.285 

6.293 

6.301 

6.309 

6.317 

122 

345 

6 

6    7 

40 

6.325 

6.332 

6.340 

6.348 

6.356 

6.364 

6.372 

6.380 

6.387 

6.395 

122 

345 

6 

6    7 

41 

6.403 

6.411 

6.419 

6.427 

6.434 

6.442 

6.450 

6.458 

6.465 

6.473 

122 

345 

5 

6    7 

42 

6.481 

6.488 

6.496 

6.504 

6.512 

6.519 

6.527 

6.535 

6.542 

6.550 

122 

345 

5 

6    7 

43 

6.557 

6.565 

6.573 

6.580 

6.588 

6.595 

6.603 

6.611 

6.618 

6.626 

122 

345 

5 

6    7 

44 

6.633 

6.641 

6.648 

6.656 

6.663 

6.671 

6.678 

6.686 

6.693 

6.701 

122 

345 

5 

6    7 

45 

6.708 

6.716 

6.723 

6.731 

6.738 

6.745 

6.753 

6.760 

6.768 

6.775 

112 

344 

5 

6    7 

46 

6.782 

6.790 

6.797 

6.804 

6.812 

6.819 

6.826 

6.834 

6.841 

6.848 

112 

344 

5 

6    7 

47 

6.856 

6.863 

6.870 

6.877 

6.885 

6.892 

6.899 

6.907 

6.914 

6.921 

112 

344 

5 

6    7 

48 

6.928 

6.935 

6.943 

6.950 

6.957 

6.964 

6.971 

6.979 

6.986 

6.993 

112 

344 

5 

6    6 

49 

7.000 

7.007 

7.014 

7.021 

7.029 

7.036 

7.043 

7.050 

7.057 

7.064 

112 

344 

5 

6    6 

50 

7.071 

7.078 

7.085 

7.092 

7.099 

7.106 

7.113 

7.120 

7.127 

7.134 

112 

344 

5 

6    6 

51 

7.141 

7.148 

7.155 

7.162 

7.169 

7.176 

7.183 

7.190 

7.197 

7.204 

112 

344 

5 

6    6 

52 

7.211 

7.218 

7.225 

7.232 

7.239 

7.246 

7.253 

7.259 

7.266 

7.273 

112 

334 

5 

6    6 

53 

7.280 

7.287 

7.294 

7.301  7.308 

7.314 

7.321 

7.328 

7.335 

7.342 

112 

334 

5 

5    6 

54 

7.348 

7.355 

7.362 

7.369  7.376 

7.382 

7.389 

7.396 

7.403 

7.409 

112 

334 

5 

5    6 

TABLE  OF  SQUAEE  EOOTS 


111 


SQUARE  ROOTS  OF  NUMBERS  FROM  10  TO  99.99 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

123 

456 

789 

55 

7.416 

7.423 

7.430 

7.436 

7.443 

7.450 

7.457 

7.463 

7.470 

7.477 

112 

334 

556 

56 

7.483 

7.490 

7.497 

7.503 

7.510 

7.517 

7.523 

7.530 

7.537 

7.543 

112 

334 

556 

57 

7.550 

7.556 

7.563 

7.570 

7.576 

7.583 

7.589 

7.596 

7.603 

7.609 

112 

334 

556 

58 

7.616 

7.622 

7.629 

7.635 

7.642 

7.649 

7.655 

7.662 

7.668 

7.675 

112 

334 

556 

59 

7.681 

7.688 

7.694 

7.701 

7.707 

7.714 

7.720 

7.727 

7.733 

7.740 

112 

334 

456 

60 

7.746 

7.752 

7.759 

7.765 

7.772 

7.778 

7.785 

7.791 

7.797 

7.804 

112 

334 

456 

61 

7.810 

7.817 

7.823 

7.829 

7.836 

7.842 

7.849 

7.855 

7.861 

7.868 

112 

334 

456 

62 

7.874 

7.880 

7.887 

7.893 

7.899 

7.906 

7.912 

7.918 

7.925 

7.931 

112 

334 

456 

63 

7.937 

7.944 

7.950 

7.956 

7.962 

7.969 

7.975 

7.981 

7.987 

7.994 

112 

334 

456 

64 

8.000 

8.006 

8.012 

8.019 

8.025 

8.031 

8.037 

8.044 

8.050 

8.056 

112 

234 

456 

65 

8.062 

8.068 

8.075 

8.081 

8.087 

8.093 

8.099 

8.106 

8.112 

8.118 

112 

234 

455 

66 

8.124 

8.130 

8.136 

8.142 

8.149 

8.155 

8.161 

8.167 

8.173 

8.179 

112 

234 

455 

67 

8.185 

8.191 

8.198 

8.204 

8.210 

8.216 

8.222 

8.228 

8.234 

8.240 

112 

234 

455 

68 

8.246 

8.252 

8.258 

8.264 

8.270 

8.276 

8.283 

8.289 

8.295 

8.301 

112 

234 

455 

69 

8.307 

8.313 

8.319 

8.325 

8.331 

8.337 

8.343 

8.349 

8.355 

8.361 

112 

234 

455 

70 

8.367 

8.373 

8.379 

8.385 

8.390 

8.396 

8.402 

8.408 

8.414 

8.420 

112 

234 

455 

71 

8.426 

8.432 

8.438 

8.444 

8.450 

8.456 

8.462 

8.468 

8.473 

8.479 

112 

234 

455 

72 

8.485 

8.491 

8.497 

8.503 

8.509 

8.515 

8.521 

8.526 

8.532 

8.538 

112 

233 

455 

73 

8.544 

8.550 

8.556 

8.562 

8.567 

8.573 

8.579 

8.585 

8.591 

8.597 

112 

233 

455 

74 

8.602 

8.608 

8.614 

8.620 

8.626 

8.631 

8.637 

8.643 

8.649 

8.654 

112 

233 

45  5 

75 

8.660 

8.666 

8.672 

8.678 

8.683 

8.689 

8.695 

8.701 

8.706 

8.712 

112 

233 

455 

76 

8.718 

8.724 

8.729 

8.735 

8.741 

8.746 

8.752 

8.758 

8.764 

8.769 

112 

233 

455 

77 

8.775 

8.781 

8.786 

8.792 

8.798 

8.803 

8.809 

8.815 

8.820 

8.826 

112 

233 

445 

78 

8.832 

8.837 

8.843 

8.849 

8.854 

8.860 

8.866 

8.871 

8.877 

8.883 

112 

233 

445 

79 

8.888 

8.894 

8.899 

8.905 

8.911 

8.916 

8.922 

8.927 

8.933 

8.939 

112 

233 

445 

80 

8.944 

8.950 

8.955 

8.961 

8.967 

8.972 

8.978 

8.983 

8.989 

8.994 

112 

233 

445 

81 

9.000 

9.006 

9.011 

9.017 

9.022 

9.028 

9.033 

9.039 

9.044 

9.050 

112 

233 

445 

82 

9.055 

9.061 

9.066 

9.072 

9.077 

9.083 

9.088 

9.094 

9.099 

9.105 

112 

233 

445 

83 

9.110 

9.116 

9.121 

9.127 

9.132 

9.138 

9.143 

9.149 

9.154 

9.160 

112 

233 

445 

84 

9.165 

9.171 

9.176 

9.182 

9.187 

9.192 

9.198 

9.203 

9.209 

9.214 

112 

233 

445 

^5 

9.220 

9.225 

9.230 

9.236 

9.241 

9.247 

9.252 

9.257 

9.263 

9.268 

112 

233 

445 

86 

9.274 

9.279 

9.284 

9.290 

9.295 

9.301 

9.306 

9.311 

9.317 

9.322 

112 

233 

445 

87 

9.327 

9.333 

9.338 

9.343 

9.349 

9.354 

9.359 

9.365 

9.370 

9.375 

112 

233 

445 

88 

9.381 

9.386 

9.391 

9.397 

9.402 

9.407 

9.413 

9.418 

9.423 

9.429 

112 

233 

445 

89 

9.434 

9.439 

9.445 

9.450 

9.455 

9.460 

9.466 

9.471 

9.476 

9.482 

112 

233 

445 

90 

9.487 

9.492 

9.497 

9.503 

9.508 

9.513 

9.518 

9.524 

9.529 

9.534 

112 

233 

445 

91 

9.539 

9.545 

9.550 

9.555 

9.560 

9.566 

9.571 

9.576 

9.581 

9.586 

112 

233 

445 

92 

9.592 

9.597 

9.602 

9.607 

9.612 

9.618 

9.623 

9.628 

9.633 

9.638 

112 

233 

445 

93 

9.644 

9.649 

9.654 

9.659 

9.664 

9.670 

9.675 

9.680 

9.685 

9.690 

112 

233 

445 

94 

9.695 

9.701 

9.706 

9.711 

9.716 

9.721 

9.726 

9.731 

9.737 

9.742 

112 

233 

445 

95 

9.747 

9.752 

9.757 

9.762 

9.767 

9.772 

9.778 

9.783 

9.788 

9.793 

112 

233 

445 

96 

9.798 

9.803 

9.808 

9.813 

9.818 

9.823 

9.829 

9.834 

9.839 

9.844 

112 

233 

445 

97 

9.849 

9.854 

9.859 

9.864 

9.869 

9.874 

9.879 

9.884 

9.889 

9.894 

112 

233 

445 

98 

9.899 

9.905 

9.910 

9.915 

9.920 

9.925 

9.930 

9.935 

9.940 

9.945 

Oil 

223 

344 

99 

9.950 

9.955 

9.960 

9.965 

9.970 

9.975 

9.980 

9.985 

9.990 

9.995 

Oil 

223 

344 

112 


MEKSUKATION 


POWERS  AND  ROOTS 


No. 

Squares 

Cubes 

Square 
Roots 

Cube 
Roots 

No. 

Squares 

Cubes 

Square 
Roots 

Cube 
Roots 

1 

1 

1 

1.000 

1.000 

51 

2  601 

132  651 

7.141 

3.708 

2 

4 

8 

1.414 

1.260 

52 

2  704 

140  608 

7.211 

3.733 

3 

9 

27 

1.732 

1.442 

53 

2  809 

148  877 

7.280 

3.756 

4 

16 

64 

2.000 

1.587 

54 

2  916 

157  464 

7.348 

3.780 

5 

25 

125 

2.236 

1.710 

55 

3  025 

166  375 

7.416 

3.803 

6 

36 

216 

2.449 

1.817 

56 

3  136 

175  616 

7.483 

3.826 

7 

49 

343 

2.646 

1.913 

57 

3  249 

185  193 

7.550 

3.849 

8 

64 

512 

2.828 

2.000 

58 

3  364 

195  112 

7.616 

3.871 

9 

81 

729 

3.000 

2.080 

59 

3  481 

205  379 

7.681 

3.893 

10 

100 

1000 

3.162 

2.154 

60 

3  600 

216  000 

7.746 

3.915 

11 

121 

1331 

3.317 

2.224 

61 

3  721 

226  981 

7.810 

3.936 

12 

144 

1728 

3.464 

2.289 

62 

3  844 

238  328 

7.874 

3.958 

13 

169 

2  197 

3.606 

2.351 

63 

3  969 

250  047 

7.937 

3.979 

14 

196 

2  744 

3.742 

2.410 

64 

4  096 

262  144 

8.000 

4.000 

15 

225 

3  375 

3.873 

2.466 

65 

4  225 

274  625 

8.062 

4.021 

16 

256 

4  096 

4.000 

2.520 

66 

4  356 

287  496 

8.124 

4.041 

17 

289 

4  913 

4.123 

2.571 

67 

4  489 

300  763 

8.185 

4.062 

18 

324 

5  832 

4.243 

2.621 

68 

4  624 

314  432 

8.246 

4.082 

19 

361 

6  859 

4.359 

2.668 

69 

4  761 

328  509 

8.307 

4.102 

20 

400 

8  000 

4.472 

2.714 

70 

4  900 

343  000 

8.367 

4.121 

21 

441 

9  261 

4.583 

2.759 

71 

5  041 

357  911 

8.426 

4.141 

22 

484 

10  648 

4.690 

2.802 

72 

5  184 

373  248 

8.485 

4.160 

23 

529 

12  167 

4.796 

2.844 

73 

5  329 

389  017 

8.544 

4.179 

24 

576 

13  824 

4.899 

2.884 

74 

5  476 

405  224 

8.602 

4.198 

25 

625 

15  625 

5.000 

2.924 

75 

5  625 

421  875 

8.660 

4.217 

26 

676 

17  576 

5.099 

2.962 

76 

5  776 

438  976 

8.718 

4.236 

27 

729 

19  683 

5.196 

3.000 

77 

5  929 

456  533 

8.775 

4.254 

28 

784 

21952 

5.292 

3.037 

78 

6  084 

474  552 

8.832 

4.273 

29 

841 

24  389 

5.385 

3.072 

79 

6  241 

493  039 

8.888 

4.291 

30 

900 

27  000 

5.477 

3.107 

80 

6  400 

512  000 

8.944 

4.309 

31 

961 

29  791 

5.568 

3.141 

81 

6  561 

531441 

9.000 

4.327 

32 

1024 

32  768 

5.657 

3.175 

82 

6  724 

551  368 

9.055 

4.344 

33 

1089 

35  937 

5.745 

3.208 

83 

6  889 

571  787 

9.110 

4.362 

34 

1156 

39  304 

5.831 

3.240 

84 

7  056 

592  704 

9.165 

4.380 

35 

1225 

42  875 

5.916 

3.271 

85 

7  225 

614  125 

9.220 

4.397 

36 

1296 

46  656 

6.000 

3.302 

86 

7  396 

636  056 

9.274 

4.414 

37 

1369 

50  653 

6.083 

3.332 

87 

7  569 

658  503 

9.327 

4.431 

38 

1444 

54  872 

6.164 

3.362 

88 

7  744 

681  472 

9.381 

4.448 

39 

1521 

59  319 

6.245 

3.391 

89 

7  921 

704  969 

9.434 

4.465 

40 

1600 

64  000 

6.325 

3.420 

90 

8  100 

729  000 

9.487 

4.481 

41 

1681 

68  921 

6.403 

3.448 

91 

8  281 

753  571 

9.539 

4.498 

42 

1764 

74  088 

6.481 

3.476 

92 

8  464 

778  688 

9.592 

4.514 

43 

1849 

79  507 

6.557 

3.503 

93 

8  649 

804  357 

9.644 

4.531 

44 

1936 

85  184 

6.633 

3.530 

94 

8  836 

830  584 

9.695 

4.547 

45 

2  025 

91125 

6.708 

3.557 

95 

9  025 

857  375 

9.747 

4.563 

46 

2  116 

97  336 

6.782 

3.583 

96 

9  216 

884  736 

9.798 

4.579 

47 

2  209 

103  823 

6.856 

3.609 

97 

9  409 

912  673 

9.849 

4.595 

48 

2  304 

110  592 

6.928 

3.634 

98 

9  604 

941  192 

9.899 

4.610 

49 

2  401 

117  649 

7.000 

3.659 

99 

9  801 

970  299 

9.950 

4.626 

60 

2  500 

125  000 

7.071 

3.684 

100 

10  000 

1  000  000 

10.000 

4.642 

SQUARE  ROOT 


113 


Square  Root  by  Trial.  If  we  have  no  square-root  tables  at 
hand  or  have  forgotten  the  method  given  on  pages  103  and 
104,  the  square  root  of  a  number  may  be  found  by  trial.  To 
understand  this  method  it  must  be  remembered  that  if  any 
number  is  divided  by  its  square  root,  the  result  is  equal  to  the 
square  root ;  for  example  : 

V4  =  2,     and     4-^2  =  2. 

V25  =  5,     and  25-^5  =  5. 


Illustrative  Problem.    Find  the  square  root 
of  7  by  the  trial  method. 

Since  7  is  about  halfway  between  4,  which  is  the 
square  of  2,  and  9,  which  is  the  square  of  3,  we  first 
try  2.5  as  the  square  root  of  7.    We  then  divide  7  by 
this  trial  root  and  obtain  2.8  as  the  result,  as  shown.   Since  the  result  is 
greater  than  the  number  by  which  we  divided,  2.5  is  evidently  too  small. 

We  next  try  2.65,  the  number  halfway  between  2.5  and  2.8.  Dividing  7 
by  2.65  we  obtain  2.642  as  the  result  to  the  nearest  thousandth'  as  shown. 
Since  the  result  is  now  less  than  the  num- 
ber by  which  we  divided,  2.65  is  too  large. 

We  then  take  2.646,  the  number  half- 
way between  2.65  and  2.642.  Dividing  7  by 
2.646  we  find  the  result  to  be  2.646.  Hence 
the  square  root  of  7  is  2.646. 

By  referring  to  the  table  it  will  be  seen 
that  2.646  is  there  given  as  the  square  root 
of  7  to  the  nearest  thousandth. 


The  student  will  soon  learn  to  judge 
from  the  difference  between  the  trial 
root  and  the  result  in  any  division 
what  number  is  the  most  sensible  to 
use  as  the  next  trial  root. 


2.641(2) 


2  65)7  00. 
5  30 
1  700 
1590 


1100 
1060 
400 
265 
135 


In  the  subsequent  exercises  tlie  student  may  use  any  of  the  methods 
of  finding  square  root  that  he  may  wish,  but  he  should  endeavor  to 
become  familiar  with  all  the  methods  given. 


114  MENSURATION 

Application  of  Square  Root  to  the  Circle.  From  the  formula 
for  the  area  of  a  circle,  A  =  jrr^,  it  is  seen  that  in  finding 
the  radius  r  we  have  to  find  a  square  root,  since 


"4 


The  formula  may  be  written  r  =  V^g^,  r  =  Vo.32  A,  r  =  Vo.3 183.1, 
according  to  the  degree  of  accuracy  required. 

For  example,  a  draftsman  who  is  to  draw  a  circle  repre- 
senting the  cross  section  of  an  iron  column  of  cross-section 
area  30  sq.  in.  needs  to  find  the  radius  of  the  circle.  If  the 
drawing  is  to  be  full  size,  what  radius  should  he  use  ? 


r  =  VO.3183  A  =  Vo.3183  x  30  =  V9.5490  =  3.090. 
Hence  to  the  nearest  0.01''  the  radius  he  should  use  is  3.09". 

Exercises.   Square  Root 

1.  Find  the  radius  that  a  tinsmith  should  use  in  laying 
out  a  circular  hole  for  a  pipe,  the  cross-section  area  of  which 
is  167  sq.  in. 

In  such  a  case  find,  the  radius  to  the  nearest  0.01"  and  then  express 
the  result  to  the  nearest  ^^'\ 

2.  What  must  be  the  diameter  of  a  water  pipe  in  order 
that  the  area  of  the  cross  section  shall  be  5sq.  in.? 

3.  Find  the  diameter  of  a  water  main  of  which  the  area 
of  the  cross  section  is  5.6  sq.  ft. 

4.  Find  the  diameter  of  a  cylinder  head  whose  area  is 
136sq.  in.,  and   of   one   whose   area  is   154  sq.  in. 

5.  A  tinsmith  is  required  to  make  some  cylindric  cans  to 
hold  a  gallon  (231  cu.  in.)  each  and  to  be  8''  high.  What 
radius  should  he  use  in  laying  out  the  base  ? 


REVIEW  EXERCISES  115 

Exercises.    Review 
Using  the  table,  find  the  sqUare  root  of  each  of  the  following : 

1.  8.5.  3.  235.  5.  3.345.  7.  74.8. 

2.  3.8.  4.  4.45.  6.  2.548.  8.  75.76. 

Bg  trial,  find  the  square  root  of  each  of  the  following  : 
9.  24.  11.  989.  13.  687.9.  15.  98.41. 

10.  38.  12.  787.  14.  332.4.  16.  8873. 

17.  The  foot  of  a  42-foot  ladder  is  15'  from  the  wall  of  a 
building  against  which  the  top  rests.  How  high  does  the 
ladder  reach  on  the  wall? 

18.  In  order  to  have  an  iron  pillar  capable  of  supporting  a 
certain  weight,  the  cross-section  area  must  be  72  sq.  in. 
What  radius  should  a  pattern  maker  use  in  drawing  the 
circle  for  the  pattern  ? 

19.  What  must  be  the  diameter  of  a  water  pipe  in  order 
that  the  area  of  the  cross  section  shall  be  7  sq.  in.? 

20.  What  must  be  the  diameter  of  the  piston  of  an  engine 
in  order  that  the  area  of  the  cross  section  may  be  146  sq.  in.  ? 

21.  A  tinsmith  wishes  to  make  some  cylindric  gallon  cans 
which  are  to  be  9'^  high.  What  must  be  the  area  of  the  base  ? 
What  radius  must  he  use  for  the  base  ? 

22.  A  cylindric  water  tank  is  28'  high  and  has  a  capacity  of 
38,000  cu.  ft.  What  is  the  diameter  of  the  tank  ?  What  would 
be  the  diameter  if  the  capacity  of  the  tank  were  doubled? 

23.  A  water  main  has  a  diameter  of  18".  What  is  the 
area  of  its  cross  section?  What  is  the  area  of  the  cross 
section  of  a  water  main  that  will  carry  twice  the  amount  of 
water  under  the  same  conditions  ?  What  is  the  diameter  of 
this  larger  main  ? 


116  MENSURATION 

Exercises.    Estimates  of  Areas 

1.  The  squared  paper  upon  which  the  figures  on  page  117 
are  drawn  is  ruled  with  ten  lines  to  the  inch,  and  the  scale 
of  the  figures  is  J^.  By  counting  the  squares  find  the  approx- 
imate area  of  figure  I.  Verify  the  result  by  finding  the  more 
nearly  exact  area  from  the  proper  formula. 

The  general  rule  to  be  followed  in  counting  or  rejecting  parts  of  a 
square  is  given  on  page  80.  In  each  exercise  given  below,  the  result 
found  by  counting  the  squares  should  be  verified  by  finding  the  area 
from  the  proper  formula. 

2.  Find  the  area  of  figure  II,  deducting  the  area  of  the 
small  circle. 

3.  Find  the  area  of  figure  III,  whicli  represents  seven 
eighths  of  the  area  of  a  circle. 

4.  Find  the  area  of  figure  IV,  the  length  of  each  curve 
line  being  a  fourth  of  a  circumference. 

5.  Find  the  area  of  figure  V,  the  length  of  each  curve 
line  being  either  a  fourth  or  a  half  of  a  circumference. 

6.  Find  the  area  of  figure  VI,  the  length  of  the  curve 
line  being  a  semicircumference. 

7.  Find  the  area  of  figure  VII,  deducting  the  area  of  the 
small  circle  from  that  of  the  large  one. 

8.  Find  the  area  of  figure  VIII,  the  length  of  the  curve 
line  being  a  fourth  of  a  circumference. 

Using  your  judgment  as  to  hoiv  the  curve  lines  are  drawn, 
find  the  area  of  each  of  the  following : 

9.  Figure  IX.  11.  Figure  XL 
10.  Figure  X.  12.  Figure  XII. 

13.  Find  the  area  of  figure  XIII,  deducting  the  area  of 
the  small  circle. 


ESTIMATES  OF  AREAS 


117 


7: 


^: 


i 


i 


-1 


<- 


^: 


:5: 


-ji. 


]r 


■1" 


p: 


;?5 


ee; 


:i? 


sii 


:22 


:^ 


.=^. 


\T 


SQ 


■S" 


E^E 


:^!: 


t 


is: 


:: 


>^? 


"iTlT' 


-V 


21 


:^ 


t 


=v 


::5: 


■-f. 


XD 


118  MENSURATION 

Useful  Formulas.  The  formulas  shown  in  the  blueprint  on 
page  119  are  useful  for  various  cases  in  mensuration,  and 
most  of  them  have  already  been  given.  Some  of  the  formulas 
already  given  are  here  printed  in  a  slightly  different  form. 
Those  not  already  given  may  be  used  with  the  others  in 
solving  the  exercises  which  follow.  They  represent  the  kind 
of  aids  often  given  on  blueprints  in  workshops,  and  the 
student  should  be  familiar  with  their  use. 

In  the  following  exercises  use  the  formulas  given  on 
page  119. 

Exercises.   Useful  Formulas 

1.  The  side  of  a  certain  square  is  given  as  6.8".  Find  the 
length  of  the  diagonal. 

The  side  being  given  to  the  nearest  0.1",  the  result  should  not  be 
given  to  any  higher  degree  of  accuracy,  and  similarly  in  all  such  cases. 

2.  Find  to  the  nearest  0.001  sq.  in.  the  area  of  a  rectangle 
13.725'^  long  and  6.375"  wide. 

3.  Find  to  the  nearest  0.1  sq.  in.  the  area  of  a  triangle  of 
base  14.9"  and  height  7.3". 

4.  Find  the  height  of  an  equilateral  triangle  of  side  7.2". 

5.  Find  the  base  of  an  equilateral  triangle  of  height  7.2"; 
of  an  equilateral  triangle  of  height  14.4". 

6.  Find  the  hypotenuse  of  a  right  triangle  of  sides  a  =  7.2", 
6  =  9.6";  of  a  right  triangle  of  sides  a  =14.4",  Z>=19.2". 

7.  Find  the  base  of  a  right  triangle  of  hypotenuse  9.8" 
and  height  7.2". 

8.  Find  the  height  of  a  right  triangle  of  hypotenuse  17.2" 
and  base  6.8". 

9.  Find  the  areas  of  circles  of  diameters  2.2"  and  4.4" 
respectively;  of  radii  8.8"  and  17.6"  respectively. 


USEFUL  FORMULAS 


119 


SQUARE 


RECTANGLE  TRIANGLE 


d  --  J. 414  b 
A-  A^ 


EQUILATERAL 
TRIANGLE 


RIGHT 
TRIANGLE 


CIRCLE 


Ji  -  o.ncG  A 


HEXAGON 


3.464  r^ 


r^  (.'^07  s 


^'-'-fv-- 3.1416 
ciK--^rd  ^Xl4l6d 

A-n.78r>4d^ 


OCTAGON       \CIRCULAR  RING] 


A=:i./4J6(7r-ry 

=  ;i.i4l6mfr)^~r) 


SPHERE 


PRISM 


CONE 


\r     rcd''^  _  4yrr^^ 


V^Area  of  l\ 


rjrca  or  lyase  x^s- 


120  MENSURATION 

10.  Find  the  circumferences  of  circles  of  diameters  3.3", 
6.6'\  and  9.9"  respectively. 

11.  Find  the  radii  of  circles  that  have  areas  of  10  sq.  in. 
and  20  sq.  in.  respectively. 

12.  Find  the  diameters  of  circles  of  circumferences  21"  and 
42"  respectively. 

13.  The  radius  of  the  circle  inscribed  in  the  Iiexagonal 
head  of  a  bolt  is  y'^g".    Find  the  area  of  the  head  of  the  bolt. 

In  the  hexagon  on  page  119  the  inside  circle,  of  which  r  is  the 
radius,  is  called  an  inscribed  circle ;  the  outside  circle  is  called  a  circum- 
scribed circle,  the  radius  of  which  is  equal  to  the  side  of  the  hexagon. 

14.  What  radius  must  be  used  to  circumscribe  a  circle 
about  a  hexagonal  nut  which  is  |"  on  a  side  ?  to  inscribe  a 
circle  within  the  nut  ? 

15.  A  side  of  the  octagonal  head  of  a  bolt  is  \".  Find  the 
area  of  the  head  of  the  bolt. 

16.  The  internal  and  external  radii  of  a  water  pipe  are 
respectively  9|"  and  10^.  Find  the  area  of  the  metal  in  the 
cross  section  of  the  pipe. 

17.  At  26^  per  square  inch  what  is  the  cost  of  gilding  a 
ball  on  the  top  of  the  dome  of  a  building,  the  ball  being 
4'  6"  in  diameter  and  no  allowance  being  made  for  the  base 
on  which  the  ball  is  supported. 

18.  What  is  the  weight  of  a  spherical  shell  w^hich  is  1|" 
thick,  which  has  an  external  diameter  of  14|",  and  is  made 
of  cast  iron  weighing  450  Ib./cu.  ft.? 

19.  Find  the  volume  of  a  prism  8"  long,  the  cross  section 
of  which  is  a  triangle  1"  on  each  side. 

20.  Find  the  volume  of  a  cone  7|"  high,  the  base  being 
a  circle  of  diameter  31".  What  would  be  the  volume  of  the 
cone  if  each  of  these  dimensions  were  doubled  ? 


REVIEW  EXERCISES  121 

Exercises.   Review 

1.  The  base  of  the  support  of  a  certain  shaft  is  a  square 
16.8"  on  a  side.  Find  to  the  nearest  0.1  sq.  in.  the  area  covered 
by  the  base  of  the  support. 

2.  Find  the  area  of  the  cross  section  of  a  cylindric  shaft 
7.2"  in  diameter. 

In  ■  solving  these  problems  and  those  which  occur  on  subsequent 
pages  continue  to  use  3^,  or  3.14,  for  ir,  unless  otherwise  directed.  The 
formulas  on  page  119,  which  give  3.1416,  may  be  used  in  such  practical 
work  as  may  require  a  very  high  degree  of  accuracy. 

3.  Find  the  radius  of  a  cyhndric  shaft  of  cross-section 
area  26.8  sq.  in. 

4.  By  the  aid  of  calipers  measuring  to  ^^"  the  diameter 
of  a  solid  shaft  was  found  to  be  4^^^".  Find  the  circum- 
ference of  the  shaft  and  the  area  of  the  cross  section. 

5.  Find  the  circumference  of  a  shaft  of  cross-section  area 
21  sq.  in.,  and  one  of  cross-section  area  63  sq.  in. 

6.  A  steel  plate  is  in  the  form  of  a  trapezoid,  the  two 
parallel  sides  being  respectively  23|"  and  17|".  The  per- 
pendicular distance  between  the  parallel  sides  is  16|".  Find 
to  the  nearest  \  sq.  in.  the  area  of  the  plate. 

Such  a  statement  always  refers  to  the  area  of  one  face. 

7.  If  the  steel  plate  in  Ex.  6  is  |"  thick,  how  many  cubic 
inches  does  it  contain  ? 

8.  A  solid  cast-iron  pillar  is  12'  6"  high  and  has  a  diam- 
eter of  8".  Taking  the  weight  of  cast  iron  as  450  Ib./cu.  ft., 
find  the  weight  of  the  pillar. 

9.  If  the  curve  surface  of  the  pillar  in  Ex.  8  is  to  be 
painted,  how  many  square  feet  must  be  covered  ? 

10.  The  area  of  the  curve  surface  of  a  cylinder  14'^  in 
height  is  176  sq.  in.   Find  the  circumference  and  the  diameter. 


122  MENSURATION 

Metric  System.  The  measures  that  were  formerly  used  by 
the  world  proved  to  be  so  unsystematic  and  inconvenient, 
varying  so  much  in  different  countries,  that  most  of  the 
civilized  nations  adopted  a  new  system  in  the  nineteenth 
century.  This  set  of  measures  was  devised  in  France  about 
the  year  1800  and  is  known  as  the  metric  system  and,  like 
our  system  of  money,  is  based  upon  the  scale  of  ten. 

The  use  of  the  metric  system  is  now  obligatory  in  more 
than  thirty  countries,  and  the  system  is  in  partial  use  in  most 
of  the  others.  Before  the  World  War  we  had  little  need  for 
the  metric  system  except  in  scientific  work,  where  it  is  used 
almost  universally.  Our  recent  great  expansion  of  foreign 
trade,  however,  necessitates  the  use  of  this  system  in  describ- 
ing goods  intended  for  export  and  in  making  machinery  of 
various  kinds.  Furthermore,  in  practical  mathematics  we  need 
to  know  the  principal  units  of  the  system  since  we  often  find 
them  used  in  teclmical  journals  and  handbooks.  We  read 
of  the  diameters  of  automobile  cylinders  in  millimeters,  of 
75-millimeter  guns,  and  of  such  distances  as  28  kilometers 
and  600  meters,  and  it  is  convenient  to  know  what  these 
various  metric  measures  mean. 

It  is  not  for  the  schools  to  decide  whether  the  metric 
system  will  probably  replace  our  common  system,  but  it  is 
important  that  they  should  give  to  students  some  knowledge 
of  the  nature  of  the  system. 

It  is  easy  to  understand  the  metric  system,  for  there  are 
only  six  important  prefixes  and  but  few  measures  to  learn. 
The  prefixes  and  their  meanings  are  as  follows: 

mini-      0.001  deka-  10 

centi-      0.01  hekto-        100 

deci-        0.1  kilo-         1000 

The  work  is  so  arranged  that  pages  122-130  may  be  omitted  if  desired. 


METRIC  SYSTEM  123 

Exercises.   Metric  System 

1.  A  mill  is  what  part  of  |1  ?  A  millimeter  is  what  part 
of  a  meter?  If  a  meter  is  about  40^^  a  millimeter  is  about 
what  part  of  an  inch  ?    Draw  a  line  about  a  millimeter  long. 

The  meter  is  equivalent  to  39.37'^,  but  in  the  problems  on  this  page, 
where  the  purpose  is  to  help  visualize  the  metric  measures,  the  meter 
may  be  taken  as  approximately  equivalent  to  40''. 

2.  A  cent  is  what  part  of  $1?  A  centimeter  is  what  part 
of  a  meter  ?  A  centimeter  is  about  what  part  of  an  inch  ? 
Draw  a  line  about  a  centimeter  long. 

3.  What  is  the  meaning  of  deci-?  A  decimeter  is  what 
part  of  a  meter?  A  decimeter  is  about  how  many  inches 
long  ?    Draw  a  line  about  a  decimeter  long. 

4.  A  French  "  75  "  is  a  French  gun  that  fires  a  projectile 
75  millimeters  in  diameter.  To  what  size  of  gun  in  use  in 
the  American  army  does  this  correspond  ? 

5.  What  is  the  meaning  of  kilo-  ?  A  kilometer  is  how  many 
meters  ?  Taking  a  meter  as  equivalent  to  31',  how  many  feet 
are  there  in  a  kilometer  ?  how  many  tenths  of  a  mile  ? 

6.  A  gram  is  approximately  equivalent  to  -^  J  ^  lb.,  and  so 
a  kilogram  is  equivalent  to  how  many  pounds  ? 

A  kilogram  is  more  nearly  equivalent  to  2.2  lb. 

7.  A  regiment  protected  by  a  battery  of  125-millimeter 
guns  advanced  3  kilometers,  captured  a  hill  160  meters  high, 
and  bombed  the  enemy's  trenches  with  bombs  weighing  a 
kilogram  apiece.  Write  the  sentence,  replacing  the  metric 
measures  with  their  equivalents  in  our  common  measures. 

8.  A  liter  is  equivalent  to  a  quart,  and  so  a  hektoliter  is 
equivalent  to  how  many  quarts?  how  many  gallons? 

The  word  liter  is  pronounced  "  lee-ter." 

By  answering  the  questions  in  these  exercises  the  student  will  learn 
some  of  the  more  important  measures  of  the  metric  system. 


124  MENSURATION 

Metric  Length.    The  table  of  metric  length  is  as  follows: 
A  kilometer  (km.)  =  1000  meters 
A  hektometer  =  100  meters 
A  dekameter  =  10  meters 
Meter  (m.) 
A  decimeter  (dm.)  =  0.1  of  a  meter 
A  centimeter  (cm.)  =  0.01  of  a  meter 
A  millimeter  (mm.)  =  0.001  of  a  meter 

The  meter  is  equivalent  to  39. 3 7'^  or  about  3^',  or 
a  little  over  a  yard;  the  kilometer  is  equivalent  to 
0.62  mi.  When  the  metric  system  was  invented,  the 
meter  was  intended  to  be  one  ten-millionth  of  the  dis- 
tance on  the  surface  of  the  earth  from  the  equator  to 
the  pole,  but  it  varies  slightly  from  this  standard. 

The  figure  at  the  right  is  a  10-centimeter,  or  1-decimeter,  rule. 
The  small  graduations  between  0  and  1  represent  millimeters. 

In  the  metric  system  only  those  measures  which  are 
printed  in  black  letters  in  the  tables  are  in  common  use. 

Any  one  of  these  measures  may  be  expressed  in 
terms  of  any  other  measure  by  simj^ly  moving  the  deci- 
mal point  to  the  right  or  left. 

Thus,  as  245^  =  24.5  dimes  =  $2.45, 

so  2475  mm.  =  247.5  cm.  =  24.75  dm.  =  2.475  m. 

All  the  units  of  the  system  are  derived  from  the  meter. 
Every  compound  name  is  accented  on  the  first  syllable,  as, 
for  example,  miVlimeter. 

The  school  should  be  supplied  with  a  meter  stick,  a  liter,  and  a 
cubic  centimeter,  and  these  can  easily  be  made  if  necessary. 

The  abbreviations  in  this  book  are  in  common  use.  Some,  however, 
use  Km.,  Dm.,  and  dm.  for  kilometer,  dekameter,  and  decimeter. 


o 

a> 

oo 

»>. 

« 

\n 

•^ 

CO 

»i 

*"" 

E 

o 

E 

METRIC  LENGTH  125 

Exercises.    Metric  Length 

1.  The  distance  from  Chicago  to  New  York  by  one  route 
is  about  1500  km.  Using  0.62  mi.  as  the  equivalent  of  the 
kilometer,  express  this  distance  in  miles. 

2.  The  distance  from  New  York  to  Albany  is  229  km. 
Express  this  distance  in  miles  as  in  Ex.  1. 

3.  In  a  gymnasium  where  scientific  records  are  kept  it  is 
found  that  a  certain  boy  is  144  cm.  tall.  Using  39.37^^  as  the 
equivalent  of  the  meter,  express  this  height  in  inches ;  in 
feet  and  inches;  in  feet  and  a  decimal. 

Practically  we  work  in  the  metric  system  or  in  our  common  system, 
rarely  having  any  occasion  to  transfer  from  one  to  the  other.  As  in 
Exs.  1  and  2,  the  purpose  here  is  merely  to  visualize  the  measures. 

4.  Measure  the  length  of  this  page  to  the  nearest  0.1  cm. ; 
to  the  nearest  0.01  dm. ;  to  the  nearest  millimeter.  Measure 
the  width  of  the  printed  portion  in  the  same  way. 

If  not  provided  with  a  ruler  marked  in  millimeters,  transfer  the 
length  to  a  strip  of  paper,  and  use  the  rule  shown  on  page  124. 

5.  Measure  the  diameter  of  a  5-cent  piece  and  the  diameter 
of  a  dime,  giving  the  results  to  the  nearest  millimeter. 

6.  Draw  a  line  AX  and  from  one  end  A  mark  oE  AC  equal 
to  17  mm. ;  then  CD  equal  to  9  mm. ;  then  DE  equal  to 
26  mm.  Measure  AIJ,  write  the  result,  and  check  by  adding 
the  separate  lengths. 

7.  Make  a  fine  dot  with  ink  on  the  rim  of  a  silver  "  quarter." 
Before  the  ink  dries  roll  the  coin  along  a  piece  of  paper  until 
the  ink  has  marked  the  paper  twice,  and  draw  a  line  connect- 
ing these  two  points.  Measure  in  millimeters  this  line,  which 
is  equal  to  the  circumference  of  the  coin,  and  also  measure 
the  diameter  of  the  coin.  Check  the  length  of  the  circum- 
ference by  the  formula  given  on  page  92. 


12G  MENSURATION 

Metric  Area.    The  table  of  metric  area  is  as  follows : 
A  square  kilometer  (sq.  km.)  =  1,000,000  square  meters 
A  square  hektometer=  10,000  square  meters 
A  square  dekameter=  100  square  meters    * 
Square  meter  (sq.  m.) 
A  square  decimeter  =  0.01  of  a  square  meter 
A  square  centimeter  (sq.  cm.)  =  0.0001  of  a  square  meter 
A  square  millimeter  (sq.mm.)=  0.000001  of  a  square  meter 

There  is  uo  generally  recognized  set  of  abbreviations  for 
square  measure.  Instead  of  using  sq.  m.,  scientific  writers 
often  use  m^.,  and  similar  abbreviations  for  the  other  measures. 

In  measuring  land  a  square  dekanieter  is  called  an  are  (pronounced  ar) ; 
a  square  hektometer  is  called  a  hektare  (ha.).  The  hektare  is  equivalent 
to  2.47  A.  (acres),  or  nearly  2^  A.  The  student  will  have  no  j^resent  use 
for  these  measures,  and  so  they  need  not  be  learned. 

Metric  Volume.    The  table  of  metric  volume  is  as  follows : 

Cubic  meter  (cu.  m.) 
A  cubic  decimeter  (cu.  dm.)  =  0.001  of  a  cubic  meter 
A  cubic  centimeter  (cu.  cm.)  =  0.000001  of  a  cubic  meter 
A  cubic  millimeter  (cu.  mm.)=  0.000000001  of  a  cubic  meter 

There  is  no  generally  recognized  set  of  abbreviations  for 
cubic  measure.  Instead  of  using  cu.  m.,  scientific  writers  often 
use  m^.,  and  similar  abbreviations  for  the  other  measures. 

We  also  have  cubic  kilometers,  cubic  hektometers,  and  cubic  deka- 
meters.    These  terms,  however,  are  seldom  used. 

In  measuring  wood  a  cubic  meter  is  called  a  stere  (st.).  The  word 
stere  is  pronounced  "  stair."  The  student  will  have  no  real  use  for  this 
measure,  however,  and  so  it  will  not  be  considered  further. 

It  is  evident  that  the  above  tables  can  be  obtained  at  once  from 
the  table  of  length  by  merely  squaring  or  cubing.  They  need  not  be 
specially  learned,  since  they  can  always  be  obtained  in  this  way,  as  is 
evident  from  Exs.  2  and  3  on  page  127. 


METRIC  AREA  AND  VOLUME  127 

Exercises.    Metric  Area  and  Volume 

1.  How  many  feet  in  1  yd.  ?  How  many  square  feet  in 
1  sq.  yd.  ?  How  many  cubic  feet  in  1  cu.  yd.  ?  How  many 
cubic  feet  in  a  cube  that  is  1 1  yd.  on  an  edge  ? 

2.  How  many  centimeters  in  1  m.  ?  How  many  square 
centimeters  in  1  sq.  m.  ?  How  many  cubic  centimeters  in 
1  cu.  m.  ?  How  many  square  centimeters  in  a  square  that 
is  1.2  m.  on  a  side? 

3.  How  many  inches  in  1  yd.  ?  How  many  square  inches 
in  1  sq.  yd.  ?  How  many  square  millimeters  in  1  sq.  cm.  ? 
How  many  cubic  millimeters  in  1  cu.  cm.  ? 

4.  Find  the  area  of  a  rectangle  17  mm.  by  28  mm. ;  of  a 
parallelogram  of  base  19.2  cm.  and  height  9.7  cm. 

5.  Find  the  area  of  a  triangle,  the  base  of  which  is  5  cm. 
and  the  height  7  cm.  Express  the  result  first  as  square  centi- 
meters and  then  as  square  millimeters. 

6.  Find  the  cross-section  area  of  the  bore  of  a  French ''  75." 

This  means  the  area  of  the  cross  section  of  the  bore  of  a  gun  that 
has  an  internal  diameter  of  75  mm. 

7.  In  Ex.  6  find  the  circumference  of  the  bore. 

8.  During  the  war  an  army  advanced  on  a  straight  front 
of  22  km.  and  occupied  a  triangular  piece  of  territory  9  km. 
in  depth,  as  shown  by  the  figure.  How 
many  square  kilometers  of  territory  did  ^/^  gj 
the  army  gain  ?                                                     ^^  ^ 

22  km 

9.  In  Ex.  8,   in   order  to  grasp   the 

situation  in  terms  of  our  common  units,  express  the  lengths 
in  miles  and  find  the  area  in  square  miles. 

10.  The  specifications  for  a  piece  of  machinery  require  a 
cylinder  0.18  m.  long  and  0.2  m.  in  diameter.  Find  the 
volume  of  the  cylinder  to  the  nearest  0.1  cu.  cm. 


128  MENSURATION 

Metric  Capacity.  The  table  of  metric  capacity  is  as  follows : 

A  hektoliter(hl.)  =  100  liters 

A  dekaliter  =  10  liters 

Liter  (1.) 

A  deciliter  (dl.)  =  0.1  of  a  liter 

A  liter  is  the  volume  of  a  cube  1  dm.  on  an  edge,  and  so  it  is  pos- 
sible to  express  this  table  in  terms  of  cubic  measures.  For  this  reason 
the  centiliter  and  milliliter  are  not  often  used. 

The  liter  is  practically  equivalent  to  our  quart. 

A  liter  contains  about  61.024  cu.  in.  and  is  equivalent  to  1.0567  liquid 
quarts  or  0.908  of  a  dry  quart.    These  details  need  not  be  memorized. 

Metric  Weight.    The  table  of  metric  weight  is  as  follows 
A  metric  ton  (t.)  =  1000  kilograms 
A  kilogram  (kg.)  =  1000  grams 
Gram  (g.) 

The  quintal,  which  is  equal  to  100  kg.,  is  also  used;  but,  like 
"  dekagram  "  and  "  hektogram,"  the  term  is  not  employed  frequeatly 
enough  to  demand  our  attention.  The  meaning  of  the  words  "decigram," 
"  centigram,"  and  "  milligram  "  is  evident,  but  the  student  will  com- 
monly find  these  units  expressed  as  decimals  of  a  gram. 

A  kilogram  is  equivalent  to  2.2  lb.,  and  a  metric  ton  to 
2204.6  lb.    A  kilogram  is  usually  called  a  kilo  (ke-lo). 

A  gram  is  equivalent  to  15.432  grains,  or  about  0.035  avoirdupois 
ounces.  A  pound  is  equivalent  to  about  0.4536  kg.  Our  common  ton  is 
equivalent  to  0.907  of  a  metric  ton. 

Using  the  metric  system,  it  is  easy  to  calculate  the  weight 
of  a  given  volume  of  a  substance  if  we  know  its  specific 
gravity,  since,  for  all  practical  purposes, 

1  cu.  cm.  of  water  weighs  1  g., 
1  1.  of  water  weighs  1  kg., 
and  1  cu.  m.  of  water  weighs  1 1. 


METKIC  CAPACITY  AND  WEIGHT  129 

Exercises.   Metric  Capacity  and  Weight 

1.  A  manufacturer  has  a  demand  from  a  South  American 
country  for  cyHnders  which  shall  contain  1251.  He  may 
roughly  estimate   this   as  how  many  gallons  per  cylinder? 

2.  An  exporting  house  receives  an  order  for  25  metric 
tons  of  copper  plates.    How  many  pounds  are  ordered  ? 

3.  A  tank  3  m.  long,  1.8  m.  wide,  and  1.2  m.  deep  is  filled 
with  water.    What  is  the  weight  of  the  water  ? 

4.  A  cylindric  gas  tank  is  1.8  m.  long  and  has  a  diameter 
of  0.3  m.    Find  the  number  of  liters  in  the  cylinder. 

5.  An  exporting  house  receives  an  order  for  a  shipment 
of  liter  measures,  each  measure  to  be  a  cylinder  0.1  m.  in 
diameter.    Find  the  height  of  each  measure. 

6.  A  liter  measure  in  the  form  of  a  cylinder  is  0.2  m.  high. 
P'ind  the  diameter  of  the  cylinder. 

7.  A  manufacturer  is  required  to  supply  a  tank  that  shall 
be  3.4  m.  long,  2.8  m.  wide,  and  that  shall  have  a  capacity  of 
19,000  1.    Find  the  height  of  the  tank  to  the  nearest  0.1  m. 

8.  An  order  is  received  by  an  exporting  concern  for 
225  metric  tons  of  cotton.  Find  to  the  nearest  bale  the 
number  of  bales  of  500  lb.  each  that  will  fill  the  order. 

9.  Find  in  kilograms  the  weight  of  the  water  that  a  tank 
3  m.  X  4  m.  X  6  m.  will  hold.  Also  find  in  pounds  the  weight 
of  the  water  that  a  tank  3'  x  4^  X  6'  will  hold,  taking  621  lb. 
as  the  weight  of  1  cu.  ft.  of  water. 

10.  The  specific  gravity  of  copper  is  8.9.  Find  the  weight 
of  a  cube  of  copper  24  cm.  on  an  edge.  Using  the  weight  of 
1  cu.  ft.  of  water  as  given  in  Ex.  9,  find  the  weight  of  a  cube 
of  copper  9^'^  on  an  edge.  Compare  this  result  with  the  result 
obtained  by  converting  into  pounds  the  weight  as  found  in 
kilograms  in  the  first  part  of  the  problem. 


130  MENSURATION 

Exercises.   Review 

1.  A  square  is  to  be  made  in  which  the  diagonal  cannot 
exceed  13.2  cm.  Find  the  greatest  length  of  side  that  can  be 
allowed  in  making  the  square. 

2.  The  side  of  one  square  is  half  as  long  as  the  side  of 
another  square.  The  area  of  the  first  square  is  what  part  of 
the  area  of  the  second  ?  The  perimeter  of  the  first  square  is 
what  part  of  the  perimeter  of  the  second  ? 

3.  The  side  of  one  square  is  equal  to  the  diagonal  of 
another  square.  The  area  of  the  first  square  is  how  many 
times  the  area  of  the  second? 

4.  The  diameter  of  one  water  pipe  is  0.4  m.  and  that  of 
another  is  0.3  m.  What  must  be  the  diameter  of  a  third 
pipe  that  shall  have  the  same  carrying  capacity  as  the  two 
smaller  pipes  together? 

5.  It  is  desired  to  replace  two  steam  pipes  that  have  the 
same  length  and  the  same  diameter  by  one  pipe  which  shall 
have  the  same  length  as  each  of  the  old  pipes  and  a  radiating 
surface  equal  to  that  of  the  two  pipes  combined.  If  the 
external  diameter  of  each  of  the  old  pipes  was  50  mm.,  what 
should  be  the  external  diameter  of  the  new  pipe  ? 

6.  Find  the  weight  of  the  water  that  it  will  take  to  fill 
a  cylindric  tank  2.5  m.  in  diameter  to  a  depth  of  2.1m. 

7.  The  entire  surface  area  of  a  cube  is  1014  sq.  cm.  Find 
the  volume  of  the  cube. 

8.  The  volume  of  a  cube  is  13,824  cu.  cm.  Find  the 
entire  surface  area  of  the  cube. 

9.  A  rectangular  water  tank  8  m.  long,  6  m.  wide,  and  5  m. 
deep  is  to  be  replaced  by  a  cylindric  tank  of  the  same  depth 
and  the  same  capacity.    Find  the  diameter  of  the  new  tank. 

10.  In  Ex.  9  find  the  circumference  and  the  area  of  the  base. 


CHAPTER  IV 


TRIGONOMETRY 


Nature  of  Trigonometry.  In  practical  measurements  one  of 
the  simplest  and  most  valuable  methods  is  trigonometry.  The 
word  comes  from  the  Greek  and  means  ''  tri-angle-measure." 
Since  we  can  easily  cut  all  plane  surfaces  actually  or  approxi- 
mately into  triangles,  we  can  measure  any  plane  surface,  at 
least  approximately,  if  we  can  measure  a  triangle. 

As  an  illustration  of  the  use  of  the  triangle  in  measuring, 
suppose  that  a  4-foot  post,  PB  in  the  figure  below,  casts  a 
shadow  6'  long  at  the  same  time  that  the  shadow  of  the 
tree  TY  is  60'  long.    Find  the  height  of  the  tree. 

We  have  two  right  triangles  ABP  and  XYT  of  the  same  shape.   We 
might  consider  triangle  A  BP  as  a  small-scale  plan  of  the  large  triangle 
XYT.    We  then  have  a  proportion 
between  the  sides,  in  which 

height  of  tree  _  height  of  post 
shadow  of  tree      shadow  of  post 

That  is,  in  this  case  we  have 
10 


I 
60 


<  = 


4  X 


=  40. 


Hence  the  tree  is  40'  high. 

We  have  here  found  the  height  of  the  tree  by  indirect 
measurement;  that  is,  we  do  not  need  to  climb  the  tree  and 
measure  the  height  directly  by  a  tape  line ;  we  find  the  height 
indirectly  by  taking  certain  other  measures. 

In  such  cases  the  object  is  assumed  to  stand  on  a  horizontal  plane. 

131 


132  TRIGONOMETRY 

Tangent  of  an  Angle.  In  finding  the  height  of  the  tree  on 
page  131  we  multiplied  the  length  of  the  shadow  of  the  tree 
by  the  ratio  of  the  height  of  the  po^t  to  its  shadow ;  that  is, 
by  the  result  of  height  divided  by  shadow.  The  height  of 
the  post  is  immaterial,  for  the  ratio  of  height  to  shadow  is 
the  same  whatever  the  height  of  the  post  which  we  take. 

That  is,  in  the  right  triangle  ACB  below,  if  angle  A  remains 
the  same,  the  ratio  of  BC  to  AC  does  not  change,  whatever 
the  length  BC  may  be. 

This  ratio  of  BC  to  AC  is  called  the  tangent  of  the  angle  A, 
It  is   customary  to  write  tan  A  for  "  tangent  of  angle  .4." 

Using  this  symbol  and  designating  BC 
in  the  figure  by  a,  and  AC  hy  h,  we  have 

taaA  =  ?; 

whence  a  =  6  tan  A, 

If  we  know  tan  A  and  can  measure  b,  we  can  compute  the 
value  of  a.  If,  therefore,  we  can  find  the  tangents  of  the 
various  angles  that  we  are  likely  to  use,  we  can  find  any 
height  by  this  method.  Our  first  problem,  therefore,  is  to 
find  the  tangents  of  angles. 

Because  of  the  variety  of  its  applications,  the  tangent  is  the  most 
natural  trigonometric  function  with  which  to  introduce  the  subject. 

In  the  above  triangle  if  we  measure  the  angle  A  with  a 
protractor,  we  find  it  to  be  about  25°,  and  if  we  measure  AC 
we  find  it  to  be  1".    We  have  just  seen  that 

a  =  b  tan  A, 
or  a  =  lx  tan  25°, 

so  that  if  we  can  find  the  value  of  tan  25°  we  can  find  the 
length  of  a  without  measuring  it  directly. 

We  shall  presently  see  that  tan  25°  =  0.17,  approximately, 
so  that  the  length  of  side  a  is  approximately  0A7"^ 


TANGENTS 


183 


Finding  Tangents.  The  method  of  finding  the  tangents 
of  various  angles  depends  upon  higher  algebra,  but  we  can 
determine  approximately  the  tangent  of 
any  angle  by  the  aid  of  squared  paper. 

In  this  figure  angle  CAB  =  14°,  and 
since  the  tangent  is  BC-^AC  we  find, 
by  counting  the  spaces,  that 


tan  14° 


91 

::l2=o.25. 
10 


/ 

B 

/ 

/ 

B 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

.^ 

B 

/ 

y 

/ 

/^ 

y 

y 

y 

B 

^ 

/ 

^ 

. 

^ 

\^ 

L 

C 

A   closer  approximation    found  by  higher 
algebra  is  0.2493. 

Similarly,     tan  CAB'  =  tan  30°  =  ^  =  0.58, 

tan  CAB"  =  tan  45°  =  ^  =  1, 
tan  CAB'^'  =  tan  51°  20'  =  4?^^  =  1.25. 


and 

B'  is  read  "7j  prime 


10 
B''  is  read  "ii  second  "  ;  /r'  is  read  "jB  third." 


Table  of  Tangents.    The  following  short  table  of  tangents 
of  angles  may  be  used  with  the  exercises  on  page  134: 


Angle 

TAN 

Angle 

TAN 

Angle 

TAN 

10° 
20° 
30° 

0.1763 
.3640 
.5774 

40° 
50° 
60° 

0.8391 
1.1918 
1.7321 

70° 
80° 
90° 

2.7475 
5.6713 

oc 

The  symbol  oo  means  "  infinity  " ;  that  is,  the  tangent  of  90°  is 
infinitely  long.   This  can  easily  be  seen  by  drawing  an  angle  of  90°. 

Students  should  be  reminded  that  the  sum  of  the  angles  in  any 
triangle  is  180°,  that  the  square  on  the  hypotenuse  of  a  right  triangle 
is  equal  to  the  sum  of  the  squares  on  the  two  sides,  that  each  angle  of 
an  equilateral  triangle  is  60°,  and  that  the  angles  at  the  base  of  an 
isosceles  triangle  are  equal. 


134  trigo:n'ometry 

Practical  Use  of  the  Tangent.  An  observer  at  A  in  the 
figure  below  determines  tlie  angle  J,  made  by  sighting  at 
the  top  i?  of  a  tower,  as  40°.  With  a  tape  with  which  he  can 
measure  accurately  to  0.01',  he  measures  the  distance  from 
A  to  the  base  C  of  the  tower  and  finds  it  n 

to  be  150'.    Find  the  height  of  the  tower. 

Since  a  =  b  tan  A, 

we  have  a  =  150  tan  40° 

=  150  X  0.8391 
=  125.865.  A  b  C 

That  is,  to  the  nearest  0.01',  the  height  of  the  tower  is  125.87'. 

If  we  had  no  instruments  for  measuring  the  distance  b 
more  closely  than  to  the  nearest  foot,  we  should  give  the 
height  of  the  tower  as  126',  since  7io  result  can  be  more  nearly 
accurate  than  the  measurements  iipon  which  it  is  based. 

Exercises.   Tangents 

1.  The  captain  of  a  ship  at  S  observes  a  lighthouse  L  to 
lie  40°  south  of  an  east-and-west  line.   After  ^q^; 
the  ship  has  sailed  to  the  position  *S'',  which       N!4o 
is  10  mi.  east  of  *S',  the  lighthouse  is  seen  to 
be  directly  south  of  the  ship.    Find  how  far 
the  lighthouse  is  from  S'. 

Referring  to  the  triangle  at  the  top  of  this  par/e,  find  in  each 
case  the  value  of  a  to  four  figm^es^  given  that  : 

2.  b  =  38',  A  =  10°.  5.  b  =  105.4',  A  =  50°. 

3.  />  =  42',  ^  =  20°.  6.  ^>  =  11(3.4',  J  =  70^ 

4.  ^  =  84',  A  =  30°.  7.  b  =  42.57',  A  =  80^ 

8.  If  each  angle  of  a  triangle  is  60°  and  the  height  of  the 
triangle  is  8",  find  the.  base. 


\o 


lO 


SINES 


135 


Sine  of  an  Angle.  Suppose  that  we  know  that  the  edge  AB 
of  the  Great  Pyramid  is  609'  to  the  nearest  foot  and  that  the 
angle  DAB  is  52°,  measures  that  are  easily  taken.  If  we  knew 
the  ratio  a  -i-  c,  which  is  shown  more  clearly  in  the  second 
figure  below,  we  could  easily  find  the 
value  of  a,  the  height.    That  is,  because 


we  have 


a  =  e  X 


a  =  609  X 


The  ratio  -  is  called  the  sine  of  A,  which  is  written  sin  A. 


In  this  figure, 
whence 


sin^  = 


a=  csin^. 


In  any  right  triangle  the  sine  of  either  acute  angle  is 
the  ratio  of  the  side  opposite  the  angle  to  the  hypotenuse. 

Finding  Sines.  We  can  determine  approximately  the  sine 
of  any  angle  by  the  aid  of  squared  paper.  In  this  figure 
the  arc  AB  is  drawn  with  center  O  and  B 
radius  10.  Then  sin  20°  is  equal  to  the 
length  of  the  perpendicular  drawn  to  OA 
from  the  point  marked  20°  in  the  figure, 
divided  by  the  radius.  The  perpendic- 
ular is  about  3.4  squares  long,  and  hence 
sin  20°  is  approximately  3.4  -i-10,  or  0.34. 

Proceeding  as  in  the  above  case,  we  find  the  following  table 
of  sines  for  the  remaining  angles  shown  in  the  figure : 


Angle 

SIN 

Angle 

SIN 

Angle 

SIN 

30° 
40° 

0.50 
.64 

50° 
52° 

0.77 
.79 

60° 
70° 

0.87 
.94 

136 


TRIGONOMETRY 


Table  of  Sines.    The  following  short  table  of  sines  may  be 
used  in  solving  the  exercises  which  are  given  on  page  137 : 


Angle 

SIN 

Angle 

SIX 

Angle 

SIN 

10° 

0.1736 

40° 

0.6428 

70° 

0.9397 

20° 

.3420 

50° 

.7660 

75° 

.9659 

30° 

.5000 

52° 

.7880 

80° 

.9848 

38° 

.6157 

60° 

.8660 

90° 

1.0000 

Practical  Use  of  the  Sine.  1.  Find  a,  the  height  of  the 
Great  Pyramid  referred  to  on  page  135. 

Since  a  =  c  sin -4, 

we  have  a  =  609  sin  52° 

=  609  X  0.7880 
=  479.892. 

Since  c  was  given  to  the  nearest  foot,  we  say  that  the  height  of  the 
Great  Pyramid  is  480'  to  the  nearest  foot. 

2.  Given  that  angle  B  is  38°,  as  is  really  the  case  in  the 
Great  Pyramid,  find  5,  which  is  half  the  diagonal  of  the  base. 

Since  5  =  c  sin  B, 

we  have  h  =  609  sin  38° 

=  609  X  0.6157 
=  374.9613. 
Hence  the  length  of  h  is  375'  to  the  nearest  foot. 

Check.  We  can  check  the  approximate  results  for  a  and 
hj  above,  from  the  formula  for  the  right  triangle  A^  >_  ^2  _^  52^ 
which  in  this  case  we  may  write  (^  =  a^-^  h\ 

While  we  shall  at  first  use  as  illustrations  such  problems  of  general 
interest  as  the  one  given  above,  it  is  evident  that  the  methods  of  trigo- 
nometry may  be  used  in  industrial  problems  as  well,  and  we  shall  soon 
apply  them  in  this  way. 


SINES  137 

Exercises.   Sines 

1.  If  a  kite  string  360'  long  makes  an  angle  of  40"^  with 
the  ground,  how  high  is  the  kite  ? 

The  kite  string  would  not  be  perfectly  straight,  but  we  can  obtain 
a  very  good  approximation  by  this  method. 

2.  In   order  to  find  the  height  of  a  mound  a  string  is 
stretched  from  A  to  B,  as  shown  in  the  5 
figure,  and  AB  is  found  to  be  68'  6"  long. 
If  ZA  is  found  to  be  30°,  what  is  the  height 
of  the  mound  ?  ^ 

The  symbol  ZA  is  used  to  indicate  the  angle  at  A,  or  the  angle  CAB. 

3.  In    planning    for    a   pontoon   bridge    across   the   head 
of  a  lake  to  save  marching  through  swamp  land,  a  squad 
of  engineer  troops  sighted  from  a  point  C  d 
due  west  to  A,    as   shown   in   the   figure. 
They  then  sighted  from  C  directly  north, 
thus  laying  out  the  line  CD,    On  this  line 
they   took    a   point  B,  measured  AB,  and  found  it  to  be 
978'.    They  found  Z^  to  be  20°.    Find  the  distance  BC 

4.  When  the  36-foot  arm  OA  of  a  steel  crane  makes  an 
angle  of  70°  with  the  horizontal  line  OB,  what  is  the 
vertical  distance  AB  ? 

5.  In  Ex.  4  find  the  horizontal  distance  OB,  the 
angle  with  the  vertical  line  AB  being  20°.  Verify  the 
lengths  oi  AB  and  OB  by  the  right-triangle  formula. 

Given  an  acute  angle  and  the  hypotenuse  of  a  right  triangle 
as  follows,  find  the  side  opposite  the  acute  angle  in  each  case  : 

6.  40°,  42.3'.  9.  60°,  12  mi.  12.  70°,  16  yd. 

7.  50°,  38.7'.  10.  30°,  28  mi.  13.  80°,  350  yd. 

8.  38°,  72.1".  11.  40°,  36  mi.  14.  75°,  400  yd. 


138 


TRIGONOMETRY 


Cosine  of  an  Angle.  In  any  triangle  the  sum  of  all  the 
angles  is  180°,  so  that  in  a  right  triangle  the  sum  of  the 
acute  angles  is  90°.  Angle  B  in  the  figure  below  is  the  com- 
plement of  angle  A\  that  is,  Z^  is  equal  to  90°  minus  Z^, 
which  is  what  we  mean  when  we  speak  of  the  complement 
of  an  angle. 

The  sine  of  the  complement  of  an  angle  is  called  the  cosine 
of  the  angle,  the  syllable  "co-"  standing  for  "complement." 
We  write  cos -4.  for  "cosine  of  angled."  By  definition,  cos^ 
is  the  same  as  sin  B,  and  in  this  figure  b 


whence 


cosil  =  - ; 
c 

b=ccosA, 


Table  of  Cosines.    The  following  short  table  of  cosines  of 
angles  may  be  used  with  the  exercises  given  on  page  139: 


Angle 

cos 

Angle 

C03 

Angle 

cos 

10° 

0.9848 

40° 

0.7660 

70° 

0.3420 

15° 

.9659 

45° 

.7071 

75° 

.2588 

20° 

.9397 

50° 

.6428 

80° 

.1736 

30° 

.8660 

60° 

.5000 

90° 

.0000 

Practical  Use  of  the  Cosine.  In  shoring  up  the  wall  BC 
it  is  desired  to  use  20-foot  timbers  and  to  have  the  shores 
make  an  angle  of  45°  with  the  horizontal.  How  far  from  the 
wall  should  the  foot  A  of  each  shore  be  placed  ? 

Since  h  =  cco^A, 

we  have  AC  =20 cos 45°  c^ 

=  20  X  0.7071 
=  14.142. 

Since  to  the  nearest  foot  AC  is.  14',  the  foot  of  each  shore  should  be 
placed  approximately  14'  from  the  base  of  the  wall. 


COSINES  139 

Exercises.   Cosines 

1.  To  find  the  distance  from  MtoN  across  a  pond,  as  here 
shown,  some  students  sighted  from  If  to  iV  and  then  ran  a 
line  MO  at  right  angles  to  MN.  They 
measured  OiV,  found  it  to  be  450  yd., 
and  found  Zi\^  to  be  60°.  Find 
the  distance  MN, 

2.  A  boy  walking  along  a  straight 
road  leaves  it  at  a  point  P  and  goes 
along  a  straight  oblique  path  500^  to  a  spring  at  S.  He  then 
takes  a  path  that  is  perpendicular  to  the  road  and  reaches  the 
road  at  a  point  ft  which  is  433'  from  P.  Find  the  angle  QPS 
which  the  oblique  path  makes  with  the  road  and  find  the 
angle  at  S  which  it  makes  with  the  other  path.  Find  also  the 
length  of  the  path  SQ. 

First  draw  the  figure  freehand  while  reading  the  problem.  What 
ratio  relating  to  angle  P  can  be  found?  What  is  its  value?  What 
angle  has  this  value  for  this  ratio?    How  may  the  angle  S  be  found? 

3.  In  this  right  triangle  suppose  that  AC=CB=\  and  then 
find  the  length  of  AB.    From  this  find  the 
value  of  sin  45°;  cos  45°;  tan  45°. 

4.  A  ship  starts  from  a  port  and  sails 
N.  20°  W.  a  distance  of  32  mi.  Find  the  dis- 
tance due  north  and  the  distance  due  west 
that  the  ship  has  sailed  from  the  port. 

The  expression  "  sails  N.  20°  W."  means  that  the  ship  sails  in  a 
direction  20°  west  of  north. 

5.  A  ship  sails  N.  10°  E.  a  distance  of  48  mi.  Find  how 
far  due  north  and  also  how  far  due  east  the  ship  has  sailed. 

6.  Draw  an  equilateral  triangle  1''  on  a  side,  draw  a  per- 
pendicular from  any  vertex  to  the  opposite  side,  and  find  the 

value  of  sin  60°;  cos  60°;  tan  60°;  sin  30°;  cos  30°;  tan  30°. 
p 


140 


TRIGONOMETRY 


Cotangent  of  an  Angle.  The  tangent  of  the  complement  of 
an  angle  is  called  the  cotangent  of  the  angle.  In  this  figure, 
therefore,  the  cotangent  of  A^  written  cot  A,  is  the  same  as 
tan  2?,  or,  expressed  as  a  formula, 

b 


cot -4  = 


whence 


b  =  a  cot  A, 


Table  of  Cotangents.    The  following  short  table  of  cotan- 
gents of  angles  may  be  used  with  the  exercises  on  page  141 : 


Angle 

COT 

Angle 

COT 

Angle 

COT 

10° 

20° 
80° 

5.6713 
2.7475 
1.7321 

40° 
50° 
60° 

1.1918 

0.8391 

.5774 

70° 

80° 
90° 

0.3640 
.1763 
.0000 

The  student  should  compare  this  table  of  cotangents  with  the  table 
of  tangents  given  on  page  133. 

Practical  Use  of  the  Cotangent.  An  observer  at  0,  on  the 
top  of  a  cliff  300'  high,  sees  the  top  of  a  floating  buoy  at  S 
at  an  angle  of  depression  of  10°.  „  o 

How  far  is  the  buoy  from  F,  the 
foot  of  the  cliff  ?  ^■ 

The  angle  of  depression  is  the  angle  which  the  line  of  sight  OS  in 
this  figure  makes  with  the  horizontal  line  OH,  The  angle  of  depression, 
or  the  angle  HOS,  is  equal  to  the  angle  S. 

Since  fc  =  acot^, 

we  have  5F=300cotlO° 

=  300  X  5.6713 

=  1701.39. 
Hence  the  buoy  is  approximately  1701'  from  the  foot  of  the  cliff. 


COTANGENTS  141 

Exercises.   Cotangents 

1.  A  steel  truss  was  made  up  of  sections  like  ABC  in 
this  figure.  If  the  sections  were  equilateral  triangles  48'  on 
a  side,  find  the  height  CM  of  each  section. 

Each  angle  of  an  equilateral  triangle  is  60°.  Hence 
ZA=QO°  and  ZACAI  =  30°.  We  also  have  AM  =  24'. 
We  may  now  find  CM  in  several  ways,  as  by  using         Ago' 


tan  60°  or  by  using  cot  30°.  -^        M 

Since  CM  -^  AM  =  cot  30°,  to  what  is  CM  equal? 

2.  A  building  known  to  be  58'  high  is  observed  from 
across  a  ravine,  the  angle  of  elevation  of  the  top  being  20°. 
How  wide  is  the  ravine? 

In  practice  the  angle  of  elevation  is  usually  taken 
from  a  point  above  the  base  of  the  building,  and      ^ 
allowance  has  to  be  made  for  this  height. 

3.  How  far  from  the  foot  of  a  tree  60'  high  must  an 
observer  lie  in  order  that  he  may  see  the  top  of  the  tree 
at  an  angle  of  50°  ?    at  an  angle  of  40°  ? 

4.  The  top  i)  of  a  hill  is  known  to  be  275'  above  the  level 
of  a  lake  AB,   An  observer  at  A  finds  the  _p 
angle  of  elevation  of  D  to  be  10°.    Find           _^^tT(F — 'M^ 
the  distance  AC  shown  in  the  diagram.                             ^  ^ 

Referring  to  the  triangle  on  page  140,  find  6,  given  that : 

5.  a  =  34',  J[  =  10°.  12.  a  =  32'  4",  A  =  80°. 


6.  a  =  27',  A  =  20°.  13.  a  =  26'  4",  A  =  50°. 

7.  a  =  65',  A  =  30°.  14.  a  =  32'  5",  A  =  30°. 

8.  a  =  130',  A  =  40°.  15.  a  =  28'  6",  A  =  60°. 

9.  a  =  350',  A  =  50°.  16.  a  =  48'  9",  A  =  70°. 

10.  a  =  14.7",  A  =  60°.  17.  a  =  52'  8",  A  =  80°. 

11.  a  =  32.6',  A  =  70°.  18.  a  =  42'  7",  A  =  80°. 


142  TRIGONOMETRY 

Tables  of  Natural  Functions.  We  speak  of  sin  A,  cos  A, 
tan  A^  and  cot  A  as  functions  of  the  angle  A.  The  functions 
of  the  angles  thus  far  used  have  been  given  in  brief  tables  as 
occasion  required.  We  shall  now  show  how  to  find  these 
functions  from  complete  tables.  These  tables  are  given  on 
pages  144-151,  the  four  functions  being  given  for  every  6', 
that  is,  for  every  0.1°,  from  0°  to  90°. 

Ill  this  case  6'  means  6  minutes,  or  ^'^  of  1°,  We  therefore  have  the 
same  symbol  (')  used  to  mean  two  different  things ;  that  is,  feet  and 
minutes.  In  case  any  uncertainty  is  likely  to  arise  we  shall  hereafter 
use  the  abbreviation  Ji.  for  feet,  but  in  general  the  context  will  make 
clear  the  meaning  of  the  symbol. 

The  tables  are  called  tables  of  natural  functions  to  distinguish  them 
from  tables  of  logarithmic  functions,  which  are  used  in  more  extended 
courses  in  trigonometry. 

For  example,  to  find  sin  28°  36',  look  for  28°  in  the  column 
at  the  left  on  page  144,  and  then  to  the  right  in  that  line 
under  36'  we  find  0.4787,  which  is  the  sine  of  28°  36'. 

To  find  sin  28°  40',  however,  we  shall  have  to  use  th( 
columns  of  differences  at  the  right-hand  side  of  the  table. 

Since  28°  40'  is  4'  more  than  28°  36',  which  we  found 
above,  we  look  in  the  columns  of  differences  under  4'  and  in 
line  with  28°  and  find  the  number  10.    We  then  have 

sin  28°  36'  =  0.4787,  as  above. 

Difference  for        £  =    .0010 

Adding,  sin  28°  40'  =  0.4797 

The  above  addition  should  be  made  mentally  as  we  look 
at  sin  28°  36';  we  write  only  0.4797. 

There  is  another  way  of  finding  sin  28°  40'.  We  notice  that  40'  is  f 
of  the  way  from  36'  to  42'  and  that  the  difference  between  sin  28°  i 
and  sin  28°  42'  is  0.0015,  so  we  simply  add  |  of  0.0015  to  sin  28°  36'. 
The  result  is  the  same  as  the  one  given  above,  but  this  process,  which 
is  ordinarily  known  as  interpolation,  is  longer. 


TABLES  OF  FUNCTIONS  148 

Illustrative  Problems.    1.  Find  sin  62°  19'. 

From  page  145  sin  62°  18'  =  0.8854 

Difference  for  V  =  1 

Adding,  sin  62°  19'  =  0.8855 

As  already  stated,  in  such  cases  we  write  only  the  result. 

2.  Find  cos  39°  50'. 

In  the  case  of  the  cosine  and  cotangent  we  must  bear  in  mind  that 
these  functions  decrease  as  the  angle  increases.  To  call  attention  to  this 
fact  the  columns  of  differences  are  marked  "  —  Differences." 

From  page  146  cos  39°  48'  =  0.7683 

Difference  for  2'  =  4 

Subtracting,  cos  39°  50'  =  0.7679 

.  3.  Find  tan  77°  39'. 

Here  the  tangent  is  changing  so  rapidly  that  the  column  of  differences 
ceases  to  be  accurate  enough.  We  therefore  use  the  plan  of  interpolation 
suggested  at  the  foot  of  page  142. 

From  page  149  tan  77°  42'  =  4.5864 

tan  77°  36'  =  4.5483 
Subtracting,  the  difference  for  6'  =  0.0381 

Taking  half  of  this,  the  difference  for      3'  =  0.0191. 
Adding  to  tan  77°  36',  tan  77°  39'  =  4.5674. 

4.  Find  the  angle  of  which  the  sine  is  0.9673. 

We  look  in  the  table  for  sines  beginning  with  the  figures  96,  and  find 
on  page  145  that  0.9673  is  in  the  line  for  75°  and  under  the  column  18'. 

Therefore  the  angle  of  which  the  sine  is  0.9673  is  75°  18'. 

5.  Find  the  angle  of  which  the  cotangent  is  0.3512. 

As  in  Ex.  4,  0.3522  =  cot  70°  36'. 

But  0.3512  is  0.0010  less  than  this,  and  from  the  columns  of  differences 
0.0010  is  to  be  subtracted  for  an  increase  of  3'.    Therefore  we  have 

0.3512  =  cot  70°  39'. 
That  is,  the  angle  of  which  the  cotangent  is  0.3512  is  70°  39'. 


144 


NATURAL  SINES.    0°-45= 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences  1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

3 
3 
3 
3 
3 

2' 

6 
6 
6 
6 
6 

3' 

9 
9 
9 
9 
9 

4' 

12 
12 
12 
12 
12 

5' 

15 
15 
15 
15 
14 

0 
1 

2 
3 

4 

.0000 
.0175 
.0349 
.0523 
.0698 

.0017 
.0192 
.0366 
.0541 
.0715 

.0035 
.0209 
.0384 
.0558 
.0732 

.0052 
.0227 
.0401 
.0576 
.0750 

.0070 
.0244 
.0419 
.0593 
.0767 

.0087 
.0262 
.0436 
.0610 
.0785 

.0105 
.0279 
.0454 
.0628 
.0802 

.0122 
.0297 
.0471 
.0645 
.0819 

.0140 
.0314 
.0488 
.0663 
.0837 

.0157 
.0332 
.0506 
.0680 
.0854 

5 
6 
7 
8 
9 

.0872 
.1045 
.1219 
.1392 
.1564 

.0889 
.1063 
.1236 
.1409 
.1582 

.0906 
.1080 
.1253 
.1426 
.1599 

.0924 
.1097 
.1271 
.1444 
.1616 

.0941 
.1115 
.1288 
.1461 
.1633 

.0958 
.1132 
.1305 
.1478 
.1650 

.0976 
.1149 
.1323 
.1495 
.1668 

.0993 
.1167 
.1340 
.1513 
.1685 

.1011 
.1184 
.1357 
.1530 
.1702 

.1028 
.1201 
.1374 
.1547 
.1719 

3 
3 
3 
3 
3 

6 
6 
6 
6 
6 

9 
9 
9 
9 
9 

12 
12 
12 
12 
12 

14 
14 
14 
14 
14 

10 
11 
12 
13 
14 

.1736 
.1908 
.2079 
.2250 
.2419 

.1754 
.1925 
.2096 
.2267 
.2436 

.1771 
.1942 
.2113 
.2284 
.2453 

.1788 
.1959 
.2130 
.2300 
.2470 

.1805 
.1977 
.2147 
.2317 
.2487 

.1822 
.1994 
.2164 
.2334 
.2504 

.1840 
.2011 
.2181 
.2351 
.2521 

.1857 
.2028 
.2198 
.2368 
.2538 

.1874 
.2045 
.2215 
.2385 
.2554 

.1891 
.2062 
.2233 
.2402 
.2571 

3 
3 
3 
3 
3 

6 
6 
6 
6 
6 

9 
9 
9 
8 
8 

11 
11 

w 

11 

14 
14 
14 
14 
14 

15 
16 
17 
18 
19 

.2588 
.2756 
.2924 
.3090 
.3256 

.2605 
.2773 
.2940 
.3107 
.3272 

.2622 
.2790 
.2957 
.3123 
.3289 

.2639 
.2807 
.2974 
.3140 
.3305 

.2656 
.2823 
.2990 
.3156 
.3322 

.2672 
.2840 
.3007 
.3173 
.3338 

.2689 
.2857 
.3024 
.3190 
.3355 

.2706 
.2874 
.3040 
.3206 
.3371 

.2723 
.2890 
.3057 
.3223 
.3387 

.2740 
.2907 
.3074 
.3239 
.3404 

3 
3 
3 
3 
3 

6 
6 
6 
6 

5 

8 
8 
8 
8 
8 

11 
11 
11 
11 
11 

14 
14 
14 
14 
14 

20 
21 
22 
23 
24 

.3420 
.3584 
.3746 
.3907 
.4067 

.3437 
.3600 
.3762 
.3923 
.4083 

.3453 
.3616 
.3778 
.3939 
.4099 

.3469 
.3633 
.3795 
.3955 
.4115 

.3486 
.3649 
.3811 
.3971 
.4131 

.3502 
.3665 
.3827 
.3987 
.4147 

.3518 
.3681 
.3843 
.4003 
.4163 

.3535 
.3697 
.3859 
.4019 
.4179 

.3551 
.3714 
.3875 
.4035 
.4195 

.3567 
.3730 
.3891 
.4051 
.4210 

3 
3 
3 
3 
3 

5 

5 
5 
5 
5 

8 
8 
8 
8 
8 

11 
11 
11 
11 
11 

14 
14 
14 
14 
13 

25 
26 
27 
28 
29 

.4226 
.4384 
.4540 
.4695 
.4848 

.4242 
.4399 
.4555 
.4710 
.4863 

.4258 
.4415 
.4571 
.4726 
.4879 

.4274 
.4431 
.4586 
.4741 
.4894 

.4289 
.4446 
.4602 
.4756 
.4909 

.4305 
.4462 
.4617 
.4772 
.4924 

.4321 
.4478 
.4633 
.4787 
.4939 

.4337 
.4493 
.4648 
.4802 
.4955 

.4352 
.4509 
.4664 
.4818 
.4970 

.4368 
.4524 
.4679 
.4833 
.4985 

3 
3 
3 
3 
3 

5 
5 
5 
5 
5 

8 
8 
8 
8 
8 

11 

10 
10 
10 
10 

13 
13 
13 
13 
13 

30 
31 
32 
33 
34 

.5000 
.5150 
.5299 
.5446 
.5592 

.5015 
.5165 
.5314 
.5461 
.5606 

.5030 
.5180 
.5329 
.5476 
.5621 

.5045 
.5195 
.5344 
.5490 
.5635 

.5060 
.5210 
.5358 
.5505 
.5650 

.5075 
.5225 
.5373 
.5519 
.5664 

.5090 
.5240 
.5388 
.5534 
.5678 

.5105 
.5255 
.5402 
.5548 
.5693 

.5120 
.5270 
.5417 
.5563 
.5707 

.5135 
.5284 
.5432 
.5577 
.5721 

3 
2 
2 
2 
2 

5 
5 
5 
5 
5 

8 

7 
7 
7 

7 

10 
10 
10 
10 
10 

13 
12 
12 
12 
12 

35 
36 
37 
38 
39 

.5736 
.5878 
.6018 
.6157 
.6293 

.5750 
.5892 
.6032 
.6170 
.6307 

.5764 
.5906 
.6046 
.6184 
.6320 

.5779 
.5920 
.6060 
.6198 
.6334 

.5793 
.5934 
.6074 
.6211 
.6347 

.5807 
.5948 
.6088 
.6225 
.6361 

.5821 
.5962 
.6101 
.6239 
.6374 

.5835 
.5976 
.6115 
.6252 
.6388 

.5850 
.5990 
.6129 
.6266 
.6401 

.5864 
.6004 
.6143 
.6280 
.6414 

2 
2 
2 
2 
2 

5 
5 

5 
5 
4 

7 
7 
7 

I 

9 
9 
9 
9 
9 

12 
12 
12 
11 
11 

40 
41 
42 
43 
44 

.6428 
.6561 
.6691 
.6820 
.6947 

.6441 
.6574 
.6704 
.6833 
.6959 

.6455 
.6587 
.6717 
.6845 
.6972 

.6468 
.6600 
.6730 
.6858 
.6984 

.6481 
.6613 
.6743 
.6871 
.6997 

.6494 
.6626 
.6756 
.6884 
.7009 

.6508 
.6639 
.6769 
.6896 
.7022 

.6521 
.6652 
.6782 
.6909 
.7034 

.6534 
.6665 
.6794 
.6921 
.7046 

.6547 
.6678 
.6807 
.6934 
.7059 

2 
2 
2 
2 
2 

4 
4 
4 
4 
4 

7 
7 
6 
6 
6 

9 
9 
9 
8 
8 

11 
11 
11 
11 
10 

All  the  above  sines  are  less  than  1. 


NATURAL  SINES.    45^-90* 


145 


45 

46 
47 
48 
49 

50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

60 
61 
62 
63 
64 

65 
66 
67 
68 
69 

70 
71 
72 
73 
74 

75 
76 
77 
78 
79 

80 

81 
82 
83 
84 

85 
86 
87 
88 
89 


0.0' 


0' 


7071 
7193 
7314 
7431 
.7547 

.7660 
,7771 
7880 
7986 
8090 

8192 
8290 
8387 
8480 
8572 

8660 
8746 
8829 
8910 
8988 

9063 
9135 
9205 
9272 
9336 

9397 
9455 
9511 
9563 
9613 

9659 
9703 
9744 
9781 
9816 

9848 
9877 
9903 
9925 
9945 

9962 
9976 
9986 
9994 
9998 


o.r 


,7083 
,7206 
,7325 
,7443 
,7559 

7672 
7782 
7891 
7997 
8100 

8202 
8300 
8396 
8490 
8581 

8669 
8755 
8838 
8918 
8996 

9070 
9143 
9212 
9278 
9342 

9403 
9461 
9516 
9568 
9617 

9664 
9707 
9748 
9785 
9820 

9851 
9880 
9905 
9928 
9947 

9963 
9977 
9987 
9995 
9999 


0.2' 


12' 


.7096 
.7218 
.7337 
.7455 
.7570 

.7683 
.7793 
,7902 
,8007 
,8111 

8211 
8310 
8406 
8499 
8590 

8678 
8763 
8846 
8926 
9003 

9078 
9150 
9219 
9285 
9348 

9409 
9466 
9521 
9573 
.9622 

.9668 
.9711 
.9751 
.9789 
.9823 

.9854 
.9882 
.9907 
.9930 
.9949 

.9965 
.9978 
.9988 
.9995 
.9999 


0.3^ 


18' 


.7108 
.7230 
.7349 
.7466 
.7581 

.7694 
.7804 
.7912 
.8018 
.8121 

.8221 
.8320 
.8415 
.8508 
.8599 

.8686 
.8771 
.8854 
.8934 
.9011 

.9085 
.9157 
.9225 
.9291 
.9354 

.9415 
.9472 
.9527 
.9578 
.9627 

.9673 
.9715 
.9755 
.9792 
.9826 

.9857 
.9885 
.9910 
.9932 
.9951 

.9966 
.9979 
.9989 
.9996 
.9999 


0.4' 


24' 


.7120 
.7242 
.7361 
.7478 
.7593 

.7705 
.7815 
.7923 
.8028 
.8131 

.8231 
.8329 
.8425 
.8517 
.8607 

.8695 
.8780 
.8862 
.8942 
.9018 

.9092 
.9164 
.9232 
.9298 
.9361 

.9421 
.9478 
.9532 
.9583 
.9632 

.9677 
.9720 
.9759 
.9796 
.9829 

.9860 
.9888 
.9912 
.9934 
.9952 

.9968 
.9980 
.9990 
.9996 
.9999 


0.5' 


30' 


.7133 
.7254 
.7373 
.7490 
.7604 

.7716 
.7826 
.7934 
.8039 
.8141 

.8241 
.8339 
.8434 
,8526 
,8616 

,8704 
,8788 
,8870 
,8949 
,9026 

,9100 
,9171 
,9239 
,9304 
,9367 

,9426 
,9483 
,9537 
.9588 
.9636 

.9681 
.9724 
.9763 
.9799 
.9833 

.9863 
.9890 
.9914 
.9936 
.9954 

.9969 
.9981 
.9990 
.9997 
1.000 


0.6' 


36' 


.7145 
.7266 
.7385 
.7501 
.7615 

.7727 
.7837 
.7944 
.8049 
.8151 

.8251 
.8348 
.8443 
.8536 
.8625 

.8712 
.8796 
,8878 
,8957 
,9033 

,9107 
,9178 
,9245 
,9311 
,9373 

,9432 
,9489 
.9542 
.9593 
.9641 

.9686 
.9728 
.9767 
.9803 
.9836 

.9866 
.9893 
.9917 
.9938 
.9956 

.9971 
.9982 
.9991 
.9997 
1.000 


0.7' 


42' 


.7157 
.7278 
.7396 
.7513 
.7627 

.7738 
.7848 
.7955 
.8059 
.8161 

.8261 
.8358 
.8453 
.8545 
.8634 

.8721 
.8805 
.8886 
.8965 
.9041 

.9114 
.9184 
.9252 
.9317 
.9379 

.9438 
.9494 
.9548 
.9598 
.9646 

.9690 
.9732 
.9770 
.9806 
.9839 

.9869 
.9895 
.9919 
.9940 
.9957 

.9972 
.9983 
.9992 
.9997 
1.000 


0.8' 


48' 


.7169 
.7290 
.7408 
.7524 
.7638 

.7749 
.7859 
.7965 
.8070 
.8171 

.8271 
.8368 
.8462 
.8554 
.8643 

.8729 
.8813 
.8894 
.8973 
.9048 

.9121 
.9191 
.9259 
.9323 
.9385 

.9444 
.9500 
.9553 
.9603 
.9650 

.9694 
.9736 
.9774 
.9810 
.9842 

.9871 
.9898 
.9921 
.9942 
.9959 

.9973 
.9984 
.9993 
.9998 
1.000 


0.9° 


54' 


.7181 
.7302 
.7420 
.7536 
.7649 

.7760 
.7869 
.7976 
.8080 
.8181 

.8281 
.8377 
.8471 
.8563 
.8652 

.8738 
.8821 
.8902 
.8980 
.9056 

.9128 
.9198 
.9265 
.9330 
.9391 

.9449 
.9505 
.9558 
.9608 
.9655 

.9699 
.9740 
.9778 
.9813 
.9845 

.9874 
.9900 
.9923 
.9943 
.9960 

.9974 
.9985 
.9993 
.9998 
1.000 


+  Differences 


1'  2'  3'  4'   5 


The  precise  value  of  all  sines  except  sin  90°  is  less  than  1 , 


146 


NATURAL  COSINES.    0°-45' 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

—  Differences  1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

V 

2' 

3' 

4' 

5' 

"^ 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

.9999 

.9999 

.9999 

.9999 

0 

0 

0 

0 

0 

1 

.9998 

.9998 

.9998 

.9997 

.9997 

.9997 

.9996 

.9996 

.9995 

.9995 

0 

0 

0 

0 

0 

2 

.9994 

.9993 

.9993 

.9992 

.9991 

.9990 

.9990 

.9989 

.9988 

.9987 

0 

0 

0 

0 

0 

3 

.9986 

.9985 

.9984 

.9983 

.9982 

.9981 

.9980 

.9979 

.9978 

.9977 

0 

0 

1 

1 

4 

.9976 

.9974 

.9973 

.9972 

.9971 

.9969 

.9968 

.9966 

.9965 

.9963 

0 

0 

1 

1 

5 

.9962 

.9960 

.9959 

.9957 

.9956 

.9954 

.9952 

.9951 

.9949 

.9947 

0 

1 

1 

6 

.9945 

.9943 

.9942 

.9940 

.9938 

.9936 

.9934 

.9932 

.9930 

.9928 

0 

1 

2 

7 

.9925 

.9923 

.9921 

.9919 

.9917 

.9914 

.9912 

.9910 

.9907 

.9905 

0 

2 

2 

8 

.9903 

.9900 

.9898 

.9895 

.9893 

.9890 

.9888 

.9885 

.9882 

.9880 

0 

2 

2 

9 

.9877 

.9874 

.9871 

.9869 

.9866 

.9863 

.9860 

.9857 

.9854 

.9851 

0 

2 

2 

10 

.9848 

.9845 

.9842 

.9839 

.9836 

.9833 

.9829 

.9826 

.9823 

.9820 

2 

2 

3 

11 

.9816 

.9813 

.9810 

.9806 

.9803 

.9799 

.9796 

.9792 

.9789 

.9785 

2 

2 

3 

12 

.9781 

.9778 

.9774 

.9770 

.9767 

.9763 

.9759 

.9755 

.9751 

.9748 

2 

3 

3 

13 

.9744 

.9740 

.9736 

.9732 

.9728 

.9724 

.9720 

.9715 

.9711 

.9707 

2 

3 

3 

14 

.9703 

.9699 

.9694 

.9690 

.9686 

.9681 

.9677 

.9673 

9668 

.9664 

2 

3 

4 

15 

.9659 

.9655 

.9650 

.9646 

.9641 

9636 

.9632 

.9627 

.9622 

.9617 

2 

2 

3 

4 

16 

.9613 

.9608 

.9603 

.9598 

.9593 

.9588 

.9583 

.9578 

.9573 

.9568 

2 

2 

3 

4 

17 

.9563 

.9558 

.9553 

.9548 

.9542 

.9537 

.9532 

.9527 

.9521 

.9516 

2 

3 

4 

4 

18 

.9511 

.9505 

.9500 

.9494 

.9489 

.9483 

.9478 

.9472 

.9466 

.9461 

2 

3 

4 

5 

19 

.9455 

.9449 

.9444 

.9438 

.9432 

.9426 

.9421 

.9415 

.9409 

.9403 

2 

3 

4 

5 

20 

.9397 

.9391 

.9385 

.9379 

.9373 

.9367 

.9361 

.9354 

.9348 

.9342 

2 

3 

4 

5 

21 

.9336 

.9330 

.9323 

.9317 

.9311 

.9304 

.9298 

.9291 

.9285 

.9278 

2 

3 

4 

5 

22 

.9272 

.9265 

.9259 

.9252 

.9245 

.9239 

.9232 

.9225 

.9219 

.9212 

2 

3 

4 

6 

23 

.9205 

.9198 

.9191 

.9184 

.9178 

.9171 

.9164 

.9157 

.9150 

.9143 

2 

3 

5 

6 

24 

.9135 

.9128 

.9121 

.9114 

.9107 

.9100 

.9092 

.9085 

.9078 

.9070 

2 

4 

5 

6 

25 

.9063 

.9056 

.9048 

.9041 

.9033 

.9026 

.9018 

.9011 

.9003 

.8996 

3 

4 

5 

6 

26 

.8988 

.8980 

.8973 

.8965 

.8957 

.8949 

.8942 

.8934 

.8926 

.8918 

3 

4 

5 

6 

27 

.8910 

.8902 

.8894 

.8886 

.8878 

.8870 

.8862 

.8854 

.8846 

.8838 

3 

4 

5 

7 

28 

.8829 

.8821 

.8813 

.8805 

.8796 

.8788 

.8780 

.8771 

.8763 

.8755 

3 

4 

6 

7 

29 

.8746 

.8738 

.8729 

.8721 

.8712 

.8704 

.8695 

.8686 

.8678 

.8669 

3 

4 

6 

7 

30 

.8660 

.8652 

.8643 

.8634 

.8625 

.8616 

.8607 

.8599 

.8590 

.8581 

1 

3 

4 

6 

7 

31 

.8572 

.8563 

.8554 

.8545 

.8536 

.8526 

.8517 

.8508 

.8499 

.849C 

2 

3 

5 

6 

8 

32 

8480 

.8471 

.8462 

.8453 

.8443 

.8434 

.8425 

.8415 

.8406 

.8396 

2 

3 

5 

6 

8 

33 

.8387 

.8377 

.8368 

.8358 

.8348 

.8339 

.8329 

.8320 

.8310 

.8300 

2 

3 

5 

6 

8 

34 

.8290 

.8281 

.8271 

.8261 

.8251 

.8241 

.8231 

.8221 

.8211 

.8202 

2 

3 

5 

7 

8 

35 

.8192 

.8181 

.8171 

.8161 

.8151 

.8141 

.8131 

.8121 

.8111 

.8100 

2 

3 

5 

7 

8 

36 

.8090 

.8080 

.8070 

.8059 

.8049 

.8039 

.8028 

.8018 

.8007 

.7997 

2 

3 

5 

7 

9 

37 

.7986 

.7976 

.7965 

.7955 

.7944 

.7934 

.7923 

.7912 

.7902 

.7891 

2 

4 

5 

7 

9 

38 

.7880 

.7869 

.7859 

.7848 

.7837 

.7826 

.7815 

.7804 

.7793 

.7782 

2 

4 

5 

7 

9 

39 

.7771 

.7760 

.7749 

.7738 

.7727 

;7716 

.7705 

.7694 

.7683 

.7672 

2 

4 

6 

7 

9 

40 

7660 

.7649 

.7638 

.7627 

.7615 

.7604 

.7593 

.7581 

.7570 

.7559 

2 

4 

6 

8 

9 

41 

.7547 

.7536 

.7524 

.7513 

.7501 

.7490 

.7478 

.7466 

.7455 

.7443 

2 

4 

6 

8 

10 

42 

.7431 

7420 

.7408 

.7396 

.7385 

.7373 

.7361 

.7349 

.7337 

.7325 

2 

4 

6 

8 

10 

43 

.7314 

.7302 

.7290 

.7278 

.7266 

.7254 

.7242 

.7230 

.7218 

.7206 

2 

4 

6  8 

10 

44 

.7193 

.7181 

.7169 

.7157 

.7145 

.7133 

.7120 

.7108 

.7096 

.7083 

2 

4 

6  8 

10 

The  precise  value  of  all  cosines  except  cos  0°  is  less  than  1. 


NATURAL  COSINES.    45*'-90'* 


147 


0 

0.0° 

o.r 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

oF 

0.8° 

0.9° 

—  Differences  1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

4 
4 
4 
4 
4 

3' 

6 
6 
6 
7 
7 

4' 

8 
8 
9 
9 
9 

5' 

10 

45 
46 
47 
48 
49 

.7071 
.6947 
.6820 
.6691 
.6561 

.7059 
.6934 
.6807 
.6678 
.6547 

.7046 
.6921 
.6794 
.6665 
.6534 

.7034 
.6909 
.6782 
.6652 
.6521 

.7022 
.6896 
.6769 
.6639 
.6508 

.7009 
.6884 
.6756 
.6626 
.6494 

.6997 
.6871 
.6743 
.6613 
.6481 

.6984 
.6858 
.6730 
.6600 
.6468 

.6972 
.6845 
.6717 
.6587 
.6455 

.6959 
.6833 
.6704 
.6574 
.6441 

2 
2 
2 
2 
2 

50 
51 
52 
53 
54 

.6428 
.6293 
.6157 
.6018 
.5878 

.6414 
.6280 
.6143 
.6004 
.5864 

.6401 
.6266 
.6129 
.5990 
.5850 

.6388 
.6252 
.6115 
.5976 
,5835 

.6374 
.6239 
.6101 
.5962 
.5821 

.6361 
.6225 
.6088 
.5948 
.5807 

.6347 
.6211 
.6074 
.5934 
.5793 

.6334 
.6198 
.6060 
.5920 
.5779 

.6320 
.6184 
.6046 
.5906 
.5764 

.6307 
.6170 
.6032 
.5892 
.5750 

2 
2 
2 
2 
2 

4 
5 
5 
5 
5 

7 
7 
7 
7 
7 

9 
9 
9 
9 
9 

12 
12 
12 

55 
56 
57 
58 
59 

.5736 
.5592 
.5446 
.5299 
.5150 

.5721 
.5577 
.5432 
.528^ 
.5135 

.5707 
.5563 
.5417 
.5270 
.5120 

.5693 
.5548 
.5402 
.5255 
.5105 

.5678 
.5534 
.5388 
.5240 
.5090 

.5664 
.5519 
.5373 
.5225 
.5075 

.5650 
.5505 
.5358 
.5210 
.5060 

.5635 
.5490 
.5344 
.5195 
.5045 

.5621 
.5476 
.5329 
.5180 
.5030 

.5606 
.5461 
.5314 
.5165 
.5015 

2 
2 
2 
2 
3 

5 
5 
5 
5 
5 

7 
7 
7 
7 
8 

10 
10 
10 
10 
10 

12 
12 
12 
12 
13 

60 
61 
62 
63 
64 

.5000 
.4848 
.4695 
.4540 
.4384 

.4985 
.4833 
.4679 
.4524 
.4368 

.4970 
.4818 
.4664 
.4509 
.4352 

.4955 
.4802 
.4648 
.4493 
.4337 

.4939 
.4787 
.4633 
.4478 
.4321 

.4924 
.4772 
.4617 
.4462 
.4305 

.4909 
.4756 
.4602 
.4446 
.4289 

.4894 
.4741 
.4586 
.4431 
.4274 

.4879 
.4726 
.4571 
.4415 
.4258 

.4863 
.4710 
.4555 
.4399 
.4242 

3 
3 
3 
3 
3 

5 

5 
5 
5 
5 

8 
8 
8 
8 
8 

10 
10 
10 
10 
11 

13 
13 
13 
13 
13 

65 
66 
67 
68 
69 

.4226 
.4067 
.3907 
.3746 
.3584 

.4210 
.4051 
.3891 
.3730 
.3567 

.4195 
.4035 
.3875 
.3714 
.3551 

.4179 
.4019 
.3859 
.3697 
.3535 

.4163 
.4003 
.3843 
.3681 
.3518 

.4147 
.3987 
.3827 
.3665 
.3502 

.4131 
.3971 
.3811 
.3649 
.3486 

.4115 
.3955 
.3795 
.3633 
.3469 

.4099 
.3939 
.3778 
.3616 
.3453 

.4083 
.3923 
.3762 
.3600 

.3437 

3 
3 
3 
3 
3 

5 
5 
5 
5 
5 

8 
8 
8 
8 
8 

11 
11 
11 
11 
11 

13 
13 
13 
14 
14 

70 
71 
72 
73 
74 

.3420 
.3256 
.3090 
.2924 
.2756 

.3404 
.3239 
.3074 
.2907 
.2740 

.3387 
.3223 
.3057 
.2890 
.2723 

.3371 
.3206 
.3040 
.2874 
.2706 

.3355 
.3190 
.3024 
.2857 
.2689 

.3338 
.3173 
.3007 
.2840 
.2672 

.3322 
.3156 
.2990 
.2823 
.2656 

.3305 
.3140 
.2974 
.2807 
.2639 

.3289 
.3123 
.2957 
.2790 
.2622 

.3272 
.3107 
.2940 
.2773 
.2605 

3 
3 
3 
3 
3 

5 
6 
6 
6 
6 

8 
8 
8 
8 
8 

11 
11 
11 
11 
11 

14 
14 
14 
14 
14 

75 
76 
77 
78 
79 

.2588 
.2419 
.2250 
.2079 
.1908 

.2571 
.2402 
.2233 
.2062 
.1891 

.2554 
.2385 
.2215 
.2045 
.1874 

.2538 
.2368 
.2198 
.2028 
.1857 

.2521 
.2351 
.2181 
.2011 
.1840 

.2504 
.2334 
.2164 
.1994 
.1822 

.2487 
.2317 
.2147 
.1977 
.1805 

.2470 
.2300 
.2130 
.1959 
.1788 

.2453 
.2284 
.2113 
.1942 
.1771 

.2436 
.2267 
.2096 
.1925 
.1754 

3 
3 
3 
3 
3 

6 
6 
6 
6 
6 

8 
8 
9 
9 
9 

11 
11 
11 
11 
11 

14 
14 
14 
14 
14 

80 

81 
82 
83 
84 

.1736 
.1564 
.1392 
.1219 
.1045 

.1719 
.1547 
.1374 
.1201 
.1028 

.1702 
.1530 
.1357 
.1184 
.1011 

.1685 
.1513 
.1340 
.1167 
.0993 

.1668 
.1495 
.1323 
.1149 
.0976 

.1650 
.1478 
.1305 
.1132 
.0958 

.1633 
.1461 
.1288 
.1115 
.0941 

.1616 
.1444 
.1271 
.1097 
.0924 

.1599 
.1426 
.1253 
.1080 
.0906 

.1582 
.1409 
.1236 
.1063 
.0889 

3 
3 
3 
3 
3 

6 
6 
6 
6 
6 

9 
9 
9 
9 
9 

11 
12 
12 
12 
12 

14 
14 
14 
14 
14 

85 
86 
87 
88 
89 

.0872 
.0698 
.0523 
.0349 
.0175 

.0854 
.0680 
.0506 
.0332 
.0157 

.0837 
.0663 
.0488 
.0314 
.0140 

.0819 
.0645 
.0471 
.0297 
.0122 

.0802 
.0628 
.0454 
.0279 
.0105 

.0785 
.0610 
.0436 
.0262 
.0087 

.0767 
.0593 
.0419 
.0244 
.0070 

.0750 
.0576 
.0401 
.0227 
.0052 

.0732 
.0558 
.0384 
.0209 
.0035 

.0715 
.0541 
.0366 
.0192 
.0017 

3 
3 
3 
3 
3 

6 
6 
6 
6 
6 

9 
9 
9 
9 
9 

12 
12 
12 
12 
12 

14 
15 
15 
15 
15 

All  the  above  cosines  are  less  than  1. 


148 


NATUKAL  TANGENTS.    0°-45^ 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences  1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

3 

2' 

6 

3' 

9 

4' 
12 

5' 

15 

0 

0.0000 

.0017 

.0035 

.0052 

.0070 

.0087 

.0105 

.0122 

.0140 

.0157 

1 

0.0175 

.0192 

.0209 

.0227 

.0244 

.0262 

.0279 

.0297 

.0314 

.0332 

3 

6 

9 

12 

15 

2 

0.0349 

.0367 

.0384 

.0402 

.0419 

.0437 

.0454 

.0472 

.0489 

.0507 

3 

6 

9 

12 

15 

3 

0.0524 

.0542 

.0559 

.0577 

.0594 

.0612 

.0629 

.0647 

.0664 

.0682 

3 

6 

9 

12 

15 

4 

0.0699 

.0717 

.0734 

.0752 

.0769 

.0787 

.0805 

.0822 

.0840 

.0857 

3 

6 

9 

12 

15 

5 

0.0875 

.0892 

.0910 

.0928 

.0945 

.0963 

.0981 

.0998 

.1016 

.1033 

3 

6 

9 

12 

15 

6 

0.1051 

.1069 

.1086 

.1104 

.1122 

.1139 

.1157 

.1175 

.1192 

.1210 

3 

6 

9 

12 

15 

7 

0.1228 

.1246 

.1263 

.1281 

.1299 

.1317 

.1334 

.1352 

.1370 

.1388 

3 

6 

9 

12 

15 

8 

0.1405 

.1423 

.1441 

.1459 

.1477 

.1495 

.1512 

.1530 

.1548 

.1566 

3 

6 

9 

12 

15 

9 

0.1584 

.1602 

.1620 

.1638 

.1655 

.1673 

.1691 

.1709 

.1727 

.1745 

3 

6 

9 

12 

15 

10 

0.1763 

.1781 

.1799 

.1817 

.1835 

.1853 

.1871 

.1890 

.1908 

.1926 

3 

6 

9 

12 

15 

11 

0.1944 

.1962 

.1980 

.1998 

.2016 

.2035 

.2053 

.2071 

.2089 

.2107 

3 

6 

9 

12 

15 

12 

0.2126 

.2144 

.2162 

.2180 

.2199 

.2217 

.2235 

.2254 

.2272 

.2290 

3 

6 

9 

12 

15 

13 

0.2309 

.2327 

.2345 

.2364 

.2382 

.2401 

.2419 

.2438 

.2456 

.2475 

3 

6 

9 

12 

15 

14 

0.2493 

.2512 

.2530 

.2549 

.2568 

.2586 

.2605 

.2623 

.2642 

.2661 

3 

6 

9 

12 

16 

15 

0.2679 

.2698 

.2717 

.2736 

.2754 

.2773 

.2792 

.2811 

.2830 

.2849 

3 

6 

9 

13 

16 

16 

0.2867 

.2886 

.2905 

.2924 

.2943 

.2962 

.2981 

.3000 

.3019 

.3038 

3 

6 

9 

13 

16 

17 

0.3057 

.3076 

.3096 

.3115 

.3134 

.3153 

.3172 

.3191 

.3211 

.3230 

3 

6 

10 

13 

16 

18 

0.3249 

.3269 

.3288 

.3307 

.3327 

.3346 

.3365 

.3385 

.3404 

.3424 

3 

6 

10 

13 

16 

19 

0.3443 

.3463 

.3482 

.3502 

.3522 

.3541 

.3561 

.3581 

.3600 

.3620 

3 

7 

10 

13 

16 

20 

0.3640 

.3659 

.3679 

.3699 

.3719 

.3739 

.3759 

.3779 

.3799 

.3819 

3 

7 

10 

13 

17 

21 

0.3839 

.3859 

.3879 

.3899 

.3919 

.3939 

.3959 

.3979 

.4000 

.4020 

3 

7 

10 

13 

17 

22 

0.4040 

.4061 

.4081 

.4101 

.4122 

.4142 

.4163 

.4183 

.4204 

.4224 

3 

7 

10 

14 

17 

23 

0.4245 

.4265 

.4286 

.4307 

.4327 

.4348 

.4369 

.4390 

.4411 

.4431 

3 

7 

10 

14 

17 

24 

0.4452 

.4473 

.4494 

.4515 

.4536 

.4557 

.4578 

.4599 

.4621 

.4642 

4 

7 

11 

14 

18 

25 

0.4663 

.4684 

.4706 

.4727 

.4748 

.4770 

.4791 

.4813 

.4834 

.4856 

4 

7 

11 

14 

18 

20 

0.4877 

.4899 

.4921 

.4942 

.4964 

.4986 

.5008 

.5029 

.5051 

.5073 

4 

7 

11 

15 

18 

27 

0.5095 

.5117 

.5139 

.5161 

.5184 

.5206 

.5228 

.5250 

.5272 

.5295 

4 

7 

11 

15 

18 

28 

0.5317 

.5340 

.5362 

.5384 

.5407 

.5430 

.5452 

.5475 

.5498 

.5520 

4 

8 

11 

15 

19 

29 

0.5543 

.5566 

.5589 

.5612 

.5635 

.5658 

.5681 

.5704 

.5727 

.5750 

4 

8 

12 

15 

19 

30 

0.5774 

.5797 

.5820 

.5844 

.5867 

.5890 

.5914 

.5938 

.5961 

.5985 

4 

8 

12 

16 

20 

31 

0.6009 

.6032 

.6056 

.6080 

.6104 

.6128 

.6152 

.6176 

.6200 

.6224 

4 

8 

12 

16 

20 

32 

0.6249 

.6273 

.6297 

.6322 

.6346 

.6371 

.6395 

.6420 

.6445 

.6469 

4 

8 

12 

16 

20 

33 

0.6494 

.6519 

.6544 

.6569 

.6594 

.6619 

.6644 

.6669 

.6694 

.6720 

4 

8 

13 

17 

21 

34 

0.6745 

.6771 

.6796 

.6822 

.6847 

.6873 

.6899 

.6924 

.6950 

.6976 

4 

9 

13 

17 

21 

35 

0.7002 

.7028 

.7054 

.7080 

.7107 

.7133 

.7159 

.7186 

.7212 

.7239 

4 

9 

13 

18 

22 

36 

0.7265 

.7292 

.7319 

.7346 

.7373 

.7400 

.7427 

.7454 

.7481 

.7508 

5 

9 

14 

18 

23 

37 

0.7536 

.7563 

.7590 

.7618 

.7646 

.7673 

.7701 

.7729 

.7757 

.7785 

5 

9 

14 

18 

23 

38 

0.7813 

.7841 

.7869 

.7898 

.7926 

.7954 

.7983 

.8012 

.8040 

.8069 

5 

9 

14 

19 

24 

39 

0.8098 

.8127 

.8156 

.8185 

.8214 

.8243 

.8273 

.8302 

.8332 

.8361 

5 

10 

15 

20 

24 

40 

0.8391 

.8421 

.8451 

.8481 

.8511 

.8541 

.8571 

.8601 

.8632 

.8662 

5 

10 

15 

20 

25 

41 

0.8693 

.8724 

.8754 

.8785 

.8816 

.8847 

.8878 

.8910 

.8941 

.8972 

5 

10 

16 

21 

26 

42 

0.9004 

.9036 

.9067 

.9099 

.9131 

.9163 

.9195 

.9228 

.9260 

.9293 

5 

11 

16 

21 

27 

43 

0.9325 

.9358 

.9391 

.9424 

9457 

.9490 

.9523 

.9556 

.9590 

.9623 

6 

11 

17 

22 

28 

44 

0.9657 

.9691 

.9725 

.9759 

.9793 

.9827 

.98611.9896 

.9930 

.9965 

6 

11 

17 

23 

29 

All  tanireiits  less  than  tan  45"  are  less  than  1. 


NATURAL  TANGENTS.    45°- OO** 


149 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences   1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

6 

2' 

12 

3' 

18 

4' 

24 

5' 

30 

"45 

1.0000 

.0035 

.0070 

.0105 

.0141 

.0176 

.0212 

.0247 

.0283 

.0319 

46 

1.0355 

.0392 

.0428 

.0464 

.0501 

.0538 

.0575 

.0612 

.0649 

.0686 

6 

12 

18 

25 

31 

47 

1.0724 

.0761 

.0799 

.0837 

.0875 

.0913 

.0951 

.0990 

.1028 

.1067 

6 

13 

19 

25 

32 

48 

1.1106 

.1145 

.1184 

.1224 

.1263 

.1303 

.1343 

.1383 

.1423 

.1463 

7 

13 

20 

26 

33 

49 

1.1504 

.1544 

.1585 

.1626 

.1667 

.1708 

.1750 

.1792 

.1833 

.1875 

7 

14 

21 

28 

34 

50 

1.1918 

.1960 

.2002 

.2045 

.2088 

.2131 

.2174 

.2218 

.2261 

.2305 

7 

14 

22 

29 

36 

51 

1.2349 

.2393 

.2437 

.2482 

.2527 

.2572 

.2617 

.2662 

.2708 

.2753 

8 

15 

23 

30 

38 

52 

1.2799 

.2846 

.2892 

.2938 

.2985 

.3032 

.3079 

.3127 

.3175 

.3222 

8 

16 

24 

31 

39 

53 

1.3270 

.3319 

.3367 

.3416 

.3465 

.3514 

.3564 

.3613 

.3663 

.3713 

8 

16 

25 

33 

41 

54 

1.3764 

.3814 

.3865 

.3916 

.3968 

.4019 

.4071 

.4124 

.4176 

.4229 

9 

17 

26 

34 

43 

55 

1.4281 

.4335 

.4388 

.4442 

.4496 

.4550 

.4605 

.4659 

.4715 

.4770 

9 

18 

27 

36 

45 

56 

1.4826 

.4882 

.4938 

.4994 

.5051 

.5108 

.5166 

.5224 

.5282 

.5340 

10 

19 

29 

38 

48 

57 

1.5399 

.5458 

.5517 

.5577 

.5637 

.5697 

.5757 

.5818 

.5880 

.5941 

10 

20 

30 

40 

50 

58 

1.6003 

.6066 

.6128 

.6191 

.6255 

.6319 

.6383 

.6447 

.6512 

.6577 

11 

21 

32 

43 

53 

59 

1.6643 

.6709 

.6775 

.6842 

.6909 

.6977 

.7045 

.7113 

.7182 

.7251 

11 

23 

34 

45 

56 

60 

1.7321 

.7391 

.7461 

.7532 

.7603 

.7675 

.7747 

.7820 

.7893 

.7966 

12 

24 

36 

48 

60 

61 

1.8040 

.8115 

.8190 

.8265 

.8341 

.8418 

.8495 

.8572 

.8650 

.8728 

13 

26 

38 

51 

64 

62 

1.8807 

.8887 

.8967 

.9047 

.9128 

.9210 

.9292 

.9375 

.9458 

.9542 

14 

27 

41 

55 

68 

63 

1.9626 

.9711 

.9797 

.9883 

.9970 

.0057 

.0145 

.0233 

.0323 

.0413 

15 

29 

44 

58 

73 

64 

2.0503 

.0594 

.0686 

.0778 

.0872 

.0965 

.1060 

.1155 

.1251 

.1348 

16 

31 

47 

63 

78 

65 

2.1445 

.1543 

.1642 

.1742 

.1842 

.1943 

.2045 

.2148 

.2251 

.2355 

17 

34 

51 

68 

85 

66 

2.2460 

.2566 

.2673 

.2781 

.2889 

.2998 

.3109 

.3220 

.3332 

.3445 

18 

37 

55 

73 

92 

67 

2.3559 

.3673 

.3789 

.3906 

.4023 

.4142 

.4262 

.4383 

.4504 

.4627 

20 

40 

60 

79 

99 

68 

2.4751 

.4876 

.5002 

.5129 

.5257 

.5386 

.5517 

.5649 

.5782 

.5916 

22 

43 

65 

87 

108 

69 

2.6051 

.6187 

.6325 

.6464 

.6605 

.6746 

.6889 

.7034 

.7179 

.7326 

24 

47 

71 

95 

119 

70 

2.7475 

.7625 

.7776 

.7929 

.8083 

.8239 

.8397 

.8556 

.8716 

.8878 

26 

52 

78 

104 

130 

71 

2.9042 

.9208 

.9375 

.9544 

.9714 

.9887 

.0061 

.0237 

.0415 

.0595 

29 

58 

87 

116 

144 

72 

3.0777 

.0961 

.1146 

.1334 

.1524 

.1716 

.1910 

.2106 

.2305 

.2506 

32 

64 

96 

129 

161 

73 

3.2709 

.2914 

.3122 

.3332 

.3544 

.3759 

.3977 

.4197 

.4420 

.4646 

36 

72 

108 

144 

180 

74 

3,4874 

.5105 

.5339 

.5576 

.5816 

.6059 

.6305 

.6554 

.6806 

.7062 

41 

81 

122 

16312041 

75 

3.7321 

.7583 

.7848 

.8118 

.8391 

.8667 

.8947 

.9232 

.9520 

.9812 

76 

4.0108 

.0408 

.0713 

.1022 

.1335 

.1653 

.1976 

.2303 

.2635 

.2972 

77 

4.3315 

.3662 

.4015 

.4373 

.4737 

.5107 

.5483 

.5864 

.6252 

.6646 

78 

4.7046 

.7453 

.7867 

.8288 

.8716 

.9152 

.9594 

.0045 

.0504 

.0970 

Use  ordinary 

79 

5.1446 

.1929 

.2422 

.2924 

.3435 

.3955 

.4486 

.5026 

.5578 

.6140 

interpolation. 

80 

5.6713 

.7297 

.7894 

.8502 

.9124 

.9758 

.0405 

.1066 

.1742 

.2432 

81 

6.3138 

.3859 

.4596 

.5350 

.6122 

.6912 

.7720 

.8548 

.9395 

.0264 

82 

7.1154 

.2066 

.3002 

.3962 

.4947 

.5958 

.6996 

.8062 

.9158 

.0285 

83 

8.1443 

.2636 

.3863 

.5126 

.6427 

.7769 

.9152 

.0579 

.2052 

.3572 

84 

9.5144 

.6768 

.8448 

.0187 

.1988 

.3854 

.5789 

.7797 

.9882 

.2048 

85 

11.430 

11.66 

11.91 

12.16 

12.43 

12.71 

13.00 

13.30 

13.62 

13.95 

86 

14.301 

14.67 

15.06 

15.46 

15.89 

16.35 

16.83 

17.34 

17.89 

18.46 

87 

19.081 

19.74 

20.45 

21.20 

22.02 

22.90 

23.86 

24.90 

26.03 

27.27 

88 

28.636 

30.14 

31.82 

33.69 

35.80 

38.19 

40.92 

44.07 

47.74 

52.08 

89 

57.290 

63.66 

71.62 

81.85 

95.49 

114.6 

143.2 

191.0286.5 

573.0 

The  integral  part  of  tangents  in  heavy-face  type  is  1  greater  than  preceding  part. 


150 


NATURAL  COTANGENTS.  0°-45^ 


0.0° 


00 

57.290 
28.636 
19.081 
14.301 

11.430 
9.5144 
8.1443 
7.1154 
6.3138 

5.6713 
5.1446 
4.7046 
4.3315 
4.0108 

3.7321 
3.4874 
3.2709 
3.0777 
2.9042 

2.7475 
2.6051 
2.4751 
2.3559 
2.2460 

2.1445 
2.0503 
1.9626 
1.8807 
1.8040 

1.7321 
1.6643 
1.6003 
1.5399 
1.4826 

1.4281 
1.3764 
1.3270 
1.2799 
1.2349 

1.1918 
1.1504 
1.1106 
1.0724 
1.0355 


o.r 


573.0 
52.08 
27.27 
18.46 
13.95 

.2048 
.3572 
.0285 
.0264 
.2432 

.6140 
.0970 
.6646 
.2972 
.9812 

.7062 
.4646 
.2506 
.0595 
.8878 

.7326 
.5916 
.4627 
.3445 
.2355 

.1348 
.0413 
.9542 
.8728 
.7966 

.7251 
.6577 
.5941 
.5340 
.4770 

.4229 
.3713 
.3222 
.2753 
.2305 

.1875 
.1463 
.1067 
.0686 
.0319 


0.2° 


12' 


286.5 
47.74 
25.03 
17.89 
13.62 

.9882 

.2052 
.9158 
.9395 

.1742 

.5578 
.0504 
.6252 
.2635 
.9520 

.6806 
.4420 
.2305 
.0415 
.8716 

.7179 
.5782 
.4504 
.3332 
.2251 

.1251 
.0323 
.9458 
.8650 
.7893 

.7182 
.6512 
.5880 
.5282 
.4715 

.4176 
.3663 
.3175 
.2708 
.2261 

.1833 
.1423 
.1028 
.0649 
.0283 


0.3° 


18' 


191.0 
44.07 
24.90 
17.34 
13.30 

.7797 
.0579 
.8062 
.8548 
.1066 

.5026 
.0045 
.5864 
.2303 
.9232 

.6554 
.4197 
.2106 
.0237 
.8556 

.7034 
.5649 
.4383 
.3220 
.2148 

.1155 
.0233 
.9375 
.8572 
.7820 

.7113 
.6447 
.5818 
.5224 
.4659 

.4124 
.3613 
.3127 
.2662 
.2218 

.1792 
.1383 
.0990 
.0612 
.0247 


0.4° 


24' 


143.2 
40.92 
23.86 
16.83 
13.00 

.5789 
.9152 

.6996 
.7720 
.0405 

.4486 
.9594 

.5483 
.1976 
.8947 

.6305 
.3977 
.1910 
.0061 
.8397 

.6889 
.5517 
.4262 
.3109 
.2045 

.1060 
.0145 
.9292 
.8495 
.7747 

.7045 
.6383 
.5757 
.5166 
.4605 

.4071 
.3564 
.3079 
.2617 
.2174 

.1750 
.1343 
.0951 
.0575 
.0212 


0.5^ 


30' 


114.6 
38.19 
22.90 
16.35 
12.71 

.3854 
.7769 
.5958 
.6912 
.9758 

.3955 
.9152 
.5107 
.1653 
.8667 

.6059 
.3759 
.1716 
.9887 
.8239 

.6746 
.5386 
.4142 
.2998 
.1943 

.0965 
.0057 
.9210 
.8418 
.7675 

.6977 
.6319 
.5697 
.5108 
.4550 

.4019 
.3514 
.3032 
.2572 
.2131 

.1708 
.1303 
.0913 
.0538 
.0176 


0.6^ 


36' 


95.49 
35.80 
22.02 
15.89 
12.43 

.1988 
.6427 
.4947 
.6122 
.9124 

.3435 
.8716 
.4737 
.1335 
.8391 

.5816 
.3544 
.1524 
.9714 
.8083 

.6605 
.5257 
4023 
.2889 
.1842 

.0872 
.9970 

.9128 
.8341 
.7603 

.6909 
.6255 
.5637 
.5051 
.4496 

.3968 
.3465 
.2985 
.2527 
.2088 

.1667 
.1263 
.0875 
.0501 
.0141 


0.7' 


42' 


81.85 
33.69 
21.20 
15.46 
12.16 

.0187 
.5126 
.3962 
.5350 
.8502 

.2924 
.8288 
.4373 
.1022 
.8118 

.5576 
.3332 
.1334 
.9544 
.7929 

.6464 
.5129 
.3906 
.2781 
.1742 

.0778 
.9883 
.9047 
.8265 
.7532 

.6842 
.6191 
.5577 
.4994 
.4442 

.3916 
.3416 
.2938 
.2482 
.2045 

.1626 
.1224 
.0837 
.0464 


0.8' 


48' 


71.62 
31.82 
20.45 
15.06 
11.91 

.8448 

.3863 
.3002 
.4596 
.7894 

.2422 
.7867 
.4015 
.0713 
.7848 

.5339 
.3122 
.1146 
.9375 
.7776 

.6325 
.5002 
.3789 
.2673 
.1642 

.0686 
.9797 
.8967 
.8190 
.7461 

6775 
.6128 
.5517 
.4938 
.4388 

.3865 
.3367 
.2892 
.2437 
.2002 

.1585 
.1184 
.0799 
.0428 


0105  .0070 


0.9' 


54' 


63.66 
30.14 
19.74 
14.67 
11.66 

.6768 
.2636 
.2066 
.3859 
.7297 

.1929 
.7453 
.3662 
.0408 
.7583 

.5105 
.2914 
.0961 
.9208 
.7625 

.6187 
.4876 
.3673 
.2566 
.1543 

.0594 
.9711 
.8887 
.8115 
.7391 

.6709 
.6066 
.5458 
.4882 
.4335 

.3814 
.3319 
.2846 
.2393 
.1960 

.1544 
.1145 
.0761 
.0392 
.0035 


Dififerences 


r|2'|  3'  I  4'  !   5' 


Use  ordinary 
interpolation. 


81 

122 

72 

108 

64 

96 

58 

87 

52 

78 

47 

71 

43 

65 

40 

60 

37 

55 

34 

51 

31 

47 

29 

44 

27 

41 

26 

38 

24 

36 

23 

34 

21 

32 

20 

30 

19 

29 

18 

27 

17 

26 

16 

25 

16 

24 

15 

23 

14 

22 

14 

21 

13 

20 

13 

19 

12 

18 

12 

18 

163 
144 
129 
116 
104 


The  integral  part  of  cotangents  in  htavy-face  type  is  1  less  than  preceding  part. 


NATURAL  COTANGENTS.    4r>°-90^ 


151 


o 

0.0° 

o.r 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

- 

Differences  1 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

V 

2' 

11 

3' 

17 

4' 

23 

5' 

29 

■45 

1.0000 

.9965 

.9930 

.9896 

.9861 

.9827 

.9793 

.9759 

.9725 

.9691 

6 

46 

0.9657 

.9623 

.9590 

.9556 

.9523 

.9490 

.9457 

.9424 

.9391 

.9358 

6 

11 

17 

22 

28 

47 

0.9325 

.9293 

.9260 

.9228 

.9195 

.9163 

.9131 

.9099 

.9067 

.9036 

5 

11 

16 

21 

27 

48 

0.9004 

.8972 

.8941 

.8910 

.8878 

.8847 

.8816 

.8785 

.8754 

.8724 

5 

10 

16 

21 

26 

49 

0.8693 

.8662 

.8632 

.8601 

.8571 

.8541 

.8511 

.8481 

.8451 

.8421 

5 

10 

15 

20 

25 

50 

0.8391 

.8361 

.8332 

.8302 

.8273 

.8243 

.8214 

.8185 

.8156 

.8127 

5 

10 

15 

20 

24 

51 

0.8098 

.8069 

.8040 

.8012 

.7983 

.7954 

.7926 

.7898 

.7869 

.7841 

5 

9 

14 

19 

24 

52 

0.7813 

.7785 

.7757 

.7729 

.7701 

.7673 

.7646 

.7618 

.7590 

.7563 

5 

9 

14 

18 

23 

53 

0.7536 

.7508 

.7481 

.7454 

.7427 

.7400 

.7373 

.7346 

.7319 

.7292 

5 

9 

14 

18 

23 

54 

0.7265 

.7239 

.7212 

.7186 

.7159 

.7133 

.7107 

.7080 

.7054 

.7028 

4 

9 

13 

18 

22 

55 

0.7002 

.6976 

.6950 

.6924 

.6899 

.6873 

.6847 

.6822 

.6796 

.6771 

4 

9 

13 

17 

21 

56 

0.6745 

.6720 

.6694 

.6669 

.6644 

.6619 

.6594 

.6569 

.6544 

.6519 

4 

8 

13 

17 

21 

57 

0.6494 

.6469 

.6445 

.6420 

.6395 

.6371 

.6346 

.6322 

.6297 

.6273 

4 

8 

12 

16 

20 

58 

0.6249 

.6224 

.6200 

.6176 

.6152 

.6128 

.6104 

.6080 

.6056 

.6032 

4 

8 

12 

16 

20 

59 

0.6009 

.5985 

.5961 

.5938 

.5914 

.5890 

.5867 

.5844 

.5820 

.5797 

4 

8 

12 

16 

20 

60 

0.5774 

.5750 

.5727 

.5704 

.5681 

.5658 

.5635 

.5612 

.5589 

.5566 

4 

8 

12 

15 

19 

61 

0.5543 

.5520 

.5498 

.5475 

.5452 

.5430 

.5407 

.5384 

.5362 

.5340 

4 

8 

11 

15 

19 

62 

0.5317 

.5295 

.5272 

.5250 

.5228 

.5206 

.5184 

.5161 

.5139 

.5117 

4 

11 

15 

18 

63 

0.5095 

.5073 

.5051 

.5029 

.5008 

.4986 

.4964 

.4942 

.4921 

.4899 

4 

11 

15 

18 

64 

0.4877 

.4856 

.4834 

.4813 

.4791 

.4770 

.4748 

.4727 

.4706 

.4684 

4 

11 

14 

18 

65 

0.4663 

.4642 

.4621 

.4599 

.4578 

.4557 

.4536 

.4515 

.4494 

.4473 

4 

11 

14 

18 

66 

0.4452 

.4431 

.4411 

.4390 

.4369 

.4348 

.4327 

.4307 

.4286 

.4265 

3 

10 

14 

17 

67 

0.4245 

.4224 

.4204 

.4183 

.4163 

.4142 

.4122 

.4101 

.4081 

.4061 

3 

10 

14 

17 

68 

0.4040 

.4020 

.4000 

.3979 

.3959 

.3939 

.3919 

.3899 

.3879 

.3859 

3 

10 

13 

17 

69 

0.3839 

.3819 

.3799 

.3779 

.3759 

.3739 

.3719 

.3699 

.3679 

.3659 

3 

10 

13 

17 

70 

0.3640 

.3620 

.3600 

.3581 

.3561 

.3541 

.3522 

.3502 

.3482 

.3463 

3 

7 

10 

13 

16 

71 

0.3443 

.3424 

.3404 

.3385 

.3365 

.3346 

.3327 

.3307 

.3288 

.3269 

3 

6 

10 

13 

16 

72 

0.3249 

.3230 

.3211 

.3191 

.3172 

.3153 

.3134 

.3115 

.3096 

.3076 

3 

6 

10 

13 

16 

73 

0.3057 

.3038 

.3019 

.3000 

.2981 

.2962 

.2943 

.2924 

.2905 

.2886 

3 

6 

9 

13 

16 

74 

0.2867 

.2849 

.2830 

.2811 

.2792 

.2773 

.2754 

.2736 

.2717 

.2698 

3 

6 

9 

13 

16 

75 

0.2679 

.2661 

.2642 

.2623 

.2605 

.2586 

.2568 

.2549 

.2530 

.2512 

3 

6 

9 

12 

16 

76 

0.2493 

.2475 

.2456 

.2438 

.2419 

.2401 

.2382 

.2364 

.2345 

.2327 

3 

6 

9 

12 

15 

77 

0.2309 

.2290 

.2272 

.2254 

.2235 

.2217 

.2199 

.2180 

.2162 

.2144 

3 

6 

9 

12 

15 

78 

0.2126 

.2107 

.2089 

.2071 

.2053 

.2035 

.2016 

.1998 

.1980 

.1962 

3 

6 

9 

12 

15 

79 

0.1944 

.1926 

.1908 

.1890 

.1871 

.1853 

.1835 

.1817 

.1799 

.1781 

3 

6 

9 

12 

15 

80 

0.1763 

.1745 

.1727 

.1709 

.1691 

.1673 

.1655 

.1638 

.1620 

.1602 

3 

6 

9 

12 

15 

81 

0.1584 

.1566 

.1548 

.1530 

.1512 

.1495 

.1477 

.1459 

.1441 

.1423 

3 

6 

9 

12 

15 

82 

0.1405 

.1388 

.1370 

.1352 

.1334 

.1317 

.1299 

.1281 

.1263 

.1246 

3 

6 

9 

12 

15 

83 

0.1228 

.1210 

.1192 

.1175 

.1157 

.1139 

.1122 

.1104 

.1086 

.1069 

3 

6 

9 

12 

15 

84 

0.1051 

.1033 

.1016 

.0998 

.0981 

.0963 

.0945 

.0928 

.0910 

.0892 

3 

6 

9 

12 

IS 

85 

0.0875 

.0857 

.0840 

.0822 

.0805 

.0787 

.0769 

.0752 

.0734 

.0717 

3 

6 

9 

12 

15 

86 

0.0699 

.0682 

.0664 

.0647 

.0629 

.0612 

.0594 

.0577 

.0559 

.0542 

3 

6 

9 

12 

15 

87 

0.0524 

.0507 

.0489 

.0472 

.0454 

.0437 

.0419 

.0402 

.0384 

.0367 

3 

6 

9 

12 

15 

88 

0.0349 

.0332 

.0314 

.0297 

.0279 

.0262 

.0244 

.0227 

.0209 

.0192 

3 

6 

9 

12 

15 

89 

0.0175 

.0157 

.0140 

.0122 

.0105 

.0087 

.0070 

.0052 

.0035 

.0017 

3 

6 

9 

12 

15 

All  cotaugeuts  greater  than  cot  46°  are  less  than  1, 


152  TRIGONOMETRY 

Exercises.    Review 

1.  Find  the  side  of  the  greatest  square  that  can  be  milled 
on  the  round  shaft  shown  on  page  153. 

After  solving  by  using  sin  45°,  the  student  should  compare  the  solu- 
tion with  the  solution  by  the  formula  given  in  the  blueprint  on  page  119. 

2.  Find  the  side  of  the  square  which  has  a  diagonal  of 
0.675'^;  of  3.062'^  of  7.245'^ 

In  workshop  practice  the  diagonal  is  often  called  the  long  diameter. 

3.  Find  to  the  nearest  ■^^"  the  side  of  the  greatest  square 
that  can  be  milled  on  a  round  shaft  Ijf  in  diameter. 

4.  In  the  drawing   of   the  bay  window  find  the   angles 
which  c  makes  with  the  vertical  and  with  tlie  horizontal. 

In  referring  to  the  angle  which  a  line  makes  with  another  line  we 
always  mean  the  acute  angle  unless  otherwise  specified.  In  this  case 
first  find  either  tangent,  and  from  this  find  one  of  the  angles. 

5.  Find  the  angle  of  slope  of  the  gable  roof,  that  is,  the 
angle  which  the  rafter  makes  with  the  horizontal. 

6.  After  solving  Ex.  5  find  the  length  of  the  rafter. 
Since  sin  .1  =  a  -^  c,  it  is  evident  that  c  sin  .1  =  a  and  c  =  a  ^  sin  A, 

7.  Find  the  depth  of  the  sharp  V-thread,  given  the  pitch, 
or  distance  between  two  successive  threads,  as  shown. 

Fi7id  the  depth  of  a  sharp  y -thread.,  given  the  pitch  as  follows  : 

°-     9    •  ^-     16    •  ^"*    32     •  ^^*     16    •  ^^-     8    • 

13.  The  drill  jig  has  six  holes  evenly  spaced  on  a  circle  of 
diameter.  151".   Find  the  distance  between  the  successive  holes. 

This  means  the  straight-line  distance  between  the  centers. 

In  Ex.  13  find  the  distance  if  the  number  of  holes  is  as  follows  : 

14.  5.  15.   7.  16.   8.  17.  9.  18.  12. 


KEVIEW  EXERCISES 


153 


77 

4-0 

— i- 


f^OUND  SHArr 


Mapery 


BAY  WINDOW 


-33-6- 


6ABLC  f\OOF 


CR03S  SECTION  of 
SHAfiP  V-THf\EAD 


D/\/LL  xJIG 


154 


TRIGONOMETRY 


A     M 


C    D 


19.  In  this  piece  of  construction  work  ^C  =  14''  and  makes 
an  angle  of  30°  with  AB,  Find  the  length  of  the  brace  AB 
and  also  of  AC^  the  distance  that  the  brace  is 

set  off  from  BC. 

20.  In  the  figure  of  Ex.  19  it  is  known  that 
BE  =zCD  =  l()y  and  that  each  makes  an  angle 
of  60°  with  DK   Find  the  length  of  the  line  CF. 

21.  From  Exs.  19  and  20  find  the  area  of  the 
trapezoid  BCDE. 

22.  A  steel   bridge    has    a  truss  ADEF^   as   here   shown. 
It   is   known   that  .4i>  =  60',    FE=U\   and 
BF=1^',    Find    to    the    nearest   degree   the 
angle  of  slope  which  AF  makes  with  AD, 

Find  the  value  of  some  function  of  the  angle  A 
and  then  find  the  angle  which  comes  nearest  to  having  that  value  for 
the  function  used. 

23.  The  principle  of  a  range  finder  is  that  of  an  isosceles 
triangle.  The  eye  is  at  E,  and  an  object  C  is  reflected 
at  both  A  and  ^  to  a  prism  at  E.  The  instrument  is 
arranged  so  that  it  can  be  adjusted  to  focus  the  lines 
AC  and  BC  on  the  object  C.  If  AE  =  EB  =  10", 
find  EC  in  yards  when  ZA=ZB  =  89°  5^'. 

Practically,  the  distances  are  computed  in  this  way  when  the  instru- 
ment is  made,  and  are  read  off  in  yards  by  the  observer  on  a  scale  which 
is  mounted  inside  the  range  finder. 

24.  In  constructing  the  spire  represented  in  the  figure  below 
it  is   planned   to    have   AB  =  42^   and  FM=92'. 
What  angle  of  slope  must  the  builder  give  to  API 

25.  In  Ex.  24  find  the  length  of  AP  and  find 
the  angle  APB, 

26.  In  the  figure  of  Ex.  24  the  brace  CD  is  put 
in  38'  above  AB,    What  is  its  length  ? 


CHAPTER  V 

THE  SLIDE  RULE 

Nature  of  the  Slide  Rule.  The  slide  rule  is  an  instrument 
which  consists  of  two  rulers,  one  of  which  slides  along  the 
other.  These  rulers  are  so  marked  that  by  adding  the  num- 
bers corresponding  to  two  marks,  which  is  done  by  simply 
sliding  one  of  the  rulers  along  the  other,  we  can  readily  find 
the  product  of  two  numbers.  The  slide  rule  can  also  be 
used  for  division  and  for  finding  powers  and  roots. 

We  shall  not  attempt  to  explain  the  principle  on  which 
the  slide  rule  is  constructed,  but  we  shall  give  general  direc- 
tions for  its  use.  Unless  the  student  has  a  slide  rule  of  his 
own,  however,  and  can  actually  work  with  it  as  he  reads 
these  directions,  he  cannot  acquire  the  necessary  facility. 

Accuracy  of  the  Work.  Since  all  measurements  are  only 
approximately  accurate,  practically  we  need  to  have  the 
results  of  computation  relating  to  measurements  only  approx- 
imately accurate.  That  is,  if  we  are  able  to  make  a  certain 
measurement  accurately  to  0.01^',  no  computation  based  upon 
it  need  be  carried  beyond  0.01'',  but  the  computation  must  be 
accurate  to  that  point.  The  slide  rule  gives  only  approx- 
imate results,  but  results  that  are  accurate  within  certain 
definite  limits,  depending  upon  the  size  of  the  rule. 

The  slide  rule  is  used  very  extensively  by  mechanics  and 
engineers  and  also  affords  one  of  the  most  convenient  checks 
on  various  mathematical  operations.  Even  if  the  slide  rule 
used  is  so  small  as  to  give  only  three  figures  accurately,  this 
may  suffice  for  the  purpose  of  checking. 

P  166 


166 


THE  SLIDE  KULE 


It  is  needless  to  say  that  no  explanation  of  a  slide  rule 
given  in  a  book  is  at  all  complete.  No  one  can  learn  to 
use  a  slide  rule  without  having  one  to  work  with,  and  no 
written  explanation  is  ever  as  satisfactory  as  one  given 
by  an  instructor  with  an  instrument  at  hand.  Large  slide 
rules  that  can  be  read  across  the  room  can  be  obtained  for 
purposes  of  instruction,  but  they  are  not  necessary  if  each 
member  of  a  class  has  a  rule  of  his  own. 


Slide  Rule.    The  illustration  below  shows  a  common  and 
convenient  form  of  the  slide  rule. 

The  rule  in  the  middle  slides  along  the  two 
outer  rules.  The  glass  plate  in  the  middle, 
known  as  the  runner  or  cursor,  slides  either  way, 
a  hair  line  being  ruled  upon  it  to  facilitate  read- 
ing the  results.  Sometimes  a  magnifying  glass 
is  attached  to  the  runner  for  greater  ease  in 
reading  and  to  allow  a  higher  degree  of  accuracy 
in  setting  the  slide. 

Scales  A  and  B  are  alike.  Each  scale  is  dupli- 
cated, one  half  running  from  1  to  10,  and  the 
other  half  also  running  from  1  to  10,  for  reasons 
explained  later.  Scales  A  and  B  are  used  when, 
owing  to  the  different  graduation  of  C  and  D,  the 
latter  are  not  convenient. 

Scales  C  and  J)  are  also  alike,  but  they  are 
not  the  same  as  A  and  B.  If  you  measure  the 
distance  from  1  to  2  (not  the  tenths)  on  C  you 
will  see  that  it  is  twice  the  distance  from  1  to  2 
on  the  A  or  B  scale ;  in  fact,  the  2  on  C  is  exactly 
below  the  4  on  7^.  Hence,  on  C  and  D  it  is  easy 
to  find  the  mark  corresponding  to  125,  to  judge 
the  position  of  1255  very  closely,  and  even  to 
judge  with  a  fair  degree  of  accuracy  the  position 
of  such  a  number  as  1257. 


1*8  "'IF 


n 


cfej 


MULTIPLICATION 


157 


Multiplication.  Suppose  that  we  wish  to  multiply  one 
number  by  another,  taking  for  purposes  of  illustration  the 
simple  case  of  2  x  2.  We  place  1  on  C  exactly  over  2  on  D, 
We  then  look  for  2  on  C  and  find  that  it  is  exactly  over  4 
on  X>,  and  hence  we  see  that  4  is  the  product  of  2  and  2. 


Multiplier 

S  ,    16  I    17 ,    18 ,   19 


K    '     U    U    U    l4'lsMe'I.Ms'U 
Multiplicaud  -^ 


irii'inririnriririii 


* 


4 


Pioiluct 


Similarly,  in  the  same  figure,  the  product  of  2  x  1.5  is 
found  just  below  the  5  which  is  between  1  and  2 ;  that  is, 
just  below  1.5.  Hence  we  see  that  2  x  1.5  =  3.  We  have 
here  added  mechanically  the  length  marked  2  and  the  length 
marked  1.5,  the  resulting  length  being  marked  3. 

If  necessary,  such  terms  as  multiplier,  multiplicand,  dirisor,  dicidend, 
and  quotient  should  be  informally  explained. 

The  scheme  of  multiplication  is  seen  from  tlie  following: 


c 

Set     1 

lender  1.2 

C 

Set     1 

Under  2.8 

D 

Over  1.5 

Read    1.8 

1) 

Over  2.1 

Read    5.88 

L2xl.5  =  1.8 

2.8  X  2.1  =  5.88 

That  is,  flace  the  1  on  C  over  the  miiltiplicand  on  D  mid 
read  the  product  on  D  heloiv  the  midtiplier  on  C,  Determine 
the  position  of  the  decimal  point  hy  considering  the  numbers. 

In  setting  the  leftrhand  index,  the  figure  1,  on  C  over  the  multipli- 
cand, if  we  find  the  multiplier  off  the  rule  we  merely  move  the  slide 
to  the  left,  setting  the  right-hand  index  over  the  multiplicand.  We 
then  read  the  product  under  the  multiplier.  This  has  the  same  effect  as 
repeating  the  D  scale  under  the  C  scale  when  it  protrudes  to  the  right. 


168  THE  SLIDE  RULE 

Exercises.   Multiplication 

1.  A  cubic  foot  of  water  weighs  62.5  lb.  and  the  specific 
gravity  of  a  certain  grade  of  steel  is  8.  Find  the  weight  of 
1  cu.  ft.  of  this  grade  of  steel. 

The  problem  requires  the  multiplication  of  62.5  by  8.  Place  the 
right-hand  1  on  C  over  6.25  on  D  and  read  the  result  500  on  D  just 
below  the  8  on  C,  making  due  allowance  for  the  decimal  point. 

2.  The  specific  gravity  of  a  certam  grade  of  cast  iron  is 
7.2.    Find  the  weight  of  1  cu.  ft.  of  this  grade  of  cast  iron. 

Using  the  slide  rule,  perform  each  of  these  multiplications : 

3.  6x9.  8.  3.1  X  4.7.  13.  1.75  x  48. 

4.  6  X  90.  9.  4.5  x  3.6.  14.  1.75  x  4.8. 

5.  6  X  98.  10.  3.9  X  7.2.  15.  2.15  x  3.9. 

6.  6  X  9.8.  11.  2.7  X  8.7.  16.  3.35  x  5.68. 

7.  0.6  X  0.98.  12.  4.4  x  7.8.  17.  5.25  x  9.76. 

18.  Fmd  the  circumference  of  a  circle  of  diameter  9". 

We  have  to  find  tt  x  9.  Most  slide  rules  have  a  mark  on  the  .1  and  B 
scales  for  tt;  that  is,  for  3j.  If  the  student's  rule  is  not  marked  for  tt, 
he  should  use  3.14  as  the  value  of  the  ratio. 

19.  Find  the  circumference  of  a  circle  of  radius  6i". 

20.  An  iron  pillar  has  a  diameter  of  7.2'^  Find  the  cir- 
cumference of  the  pillar. 

21.  A  box  is  12.7''  long  and  7.9''  wide.  Find  the  area  of 
the  bottom  of  the  box. 

22.  How  many  inches  of  wire  will  be  needed  to  make 
100  rings,  each  of  which  is  2.5"  m  diameter? 

23.  By  the  aid  of  the  slide  rule  find  whether  27  X  43  or 
15^  X  84  has  the  larger  product. 

24.  As  in  Ex.  23,  compare  23.8  x  64.8  and  32.4  x  47.6. 


MULTIPLICATION 


159 


Continued  Multiplication.  By  using  the  runner  we  are  able 
to  perform  continued  multiplication  without  having  to  read 
the  intermediate  products.  For  example,  suppose  that  we 
wish  to  find  the  product  in  the  case  of  12  x  15  x  20. 


r                         f 

(^                        f 

^ 

cf   ,    P  ,  P  ,  P 

h 

'l  r 

P,  M^.f 

Z)|'""""l, 

I'U'  I.'  L' 

1.' 

n 

III      1 

'     i 

1  1  1  1  11^ 

In  the  illustration  the  minor  subdivisions  are  omitted.  We  set  the 
1  on  C  above  12  on  D  and  place  the  runner  so  that  the  hair  line  crosses 
15  on  C.  Now,  instead  of  reading  the  result,  bring  1  on  C  exactly  under 
the  hair  line,  and  under  2  on  C  read  the  result  3600  on  D. 

The  student  must  use  his  judgment  as  to  how  accurately  the  result 
may  be  given,  depending  upon  the  size  of  the  slide  rule  used. 


Exercises.    Continued  Multiplication 

1.  Find  the  volume  of  a  box  8''xl2'^xl8^ 

2.  At  |6.50  a  day,  how  much  will  16  men  earn  in  5  da.? 

3.  A  bill  of  goods  amounting  to  $2500  is  allowed  discounts 
of  8  and  10.    Find  the  net  amount. 

Remember  that  discounts  of  8  and  10  mean  discounts  of  8%,  10% 
and  that  in  this  case  we  have  to  find  the  value  of  0.9  x  0.92  x  $2500. 

4.  Find  the  cost  of  7  doz.  cylinder  priming  cups  for  an 
automobile  at  65^  each. 

We  have  to  find  the  value  of  7  x  12  x  $0.65.    The  result  will  some- 
what exceed  half  of  7  x  12,  and  hence  will  have  two  integral  places. 

5.  Find  the  cost  of  6  doz.  exhaust-pipe  flange  gaskets  for 
£in  automobile  at  190  each. 

Find  the  value  of  each  of  the  following : 

6.  4  X  5  X  6.         8.  22  X  30  X  60.         10.  37  x  26  x  75. 

7.  3  X  6  X  9.         9.  17  x  13  X 19.  11.  14  x  19  x  225. 


160  THE  SLIDE  EULE 

Division.    We  perform  division  on  the  slide  rule  by  simply 
reversing  multiplication.    Expressed  graphically  we  have: 


c 

Set  divisor 

Under  1  (right  or  left) 

D 

Over  dividend 

Read  quotient 

For  example,  suppose  that  we  wish  to  divide  25  by  2. 


2 


Divisor- 


i\hwm 


H    ■     ll'     U    Is'    I4'    Is'   WVWW^     .     .     I    .    I    I    .TTJ 
Quotient  -^  DividenJ  J 


As  shown  above,  we  place  the  divisor  2  on  C  over  the  dividend  25  on 
1)  and  read  125  on  D  under  the  left-hand  1  on  C.  Since  it  is  evident 
where  the  decimal  point  belongs,  we  write  12.5  as  the  result. 

For  example,  suppose  that  we  wish  to  divide  350  by  56. 

We  place  56  on  C  over  350  on  D,  and  under  the  right-hand  1  on  C 
read  625  on  D.  Since  we  have  350  divided  by  a  number  a  little  over 
50,  the  result  must  be  a  little  less  than  7.  Hence  we  should  place  the 
decimal  point  after  the  first  6 ;  that  is,  the  result  is  6.25. 

There  are  special  rules  for  determining  where  to  place  the  decimal 
point  in  the  quotient,  but  for  these  rules  the  student  should  consult  the 
manuals  which  come  with  most  slide  rules. 

Exercises.    Division 
Perforin  each  of  the  following  divisions  : 

1.  9^3.         7.  17.28 -f-12.  13.  3885^35. 

2.  3^9.         8.  1.728^1.2.  14.  38.85 -^  3.5. 

3.  46-!- 23.     '  9.  6250-25.  15.  6970 --41. 

4.  31 --93.      10.  312.5-1-2.5.  16.  70.11^4.1. 

5.  75^2.5.      11.  1.584-12.  17.  9300 -^  6.2. 

6.  12.5-f-2.5.     12.  15.84-1.2.  18.  5.329-f-73. 


MULTIPLICATION  AND  DIVISION  161 

Continued  Multiplication  and  Division.  In  practical  work 
we  sometimes  need  to  make  calculations  involving  continued 
multiplication  and  division,  such  as  the  following: 

425  X  67.3  X  300 
773  X  0.07 

Using  the  runner,  we  first  perform  the  continued  multiplication  of 
the  numerator,  placing  the  hair  line  over  the  final  product.  We  then 
divide  this  result  by  773,  and  divide  the  result  found  by  this  division 
by  0.07.    The  figures  of  the  final  result  are  1,  5,  8,  6. 

To  determine  the  position  of  the  decimal  point  we  simply  notice  that 
the  numerator  is  about  300  x  300,  or  90,000.  The  denominator  is  about 
0.07  X  800,  or  56.  Then  90,000  -^  56  is  about  9000  -^  6,  or  about  1500. 
Hence  we  see  that  the  result  must  be  1586. 


Exercises.    Multiplication  and  Division 

1.  The  pull  P  of  a  locomotive  is  given  by  the  formula 
P  =  d^ps/I),  where  d  is  the  diameter  of  each  cylinder  in 
inches,  p  the  pressure  of  steam  in  pounds  per  square  inch, 
s  the  length  of  the  stroke  in  inches,  and  D  the  diameter  of 
the  drive  wheels  in  inches.  Find  the  value  of  P,  given  that 
c?  =  10.5,  jt?  =  140,  8=18,  andD=36. 

2.  For  a  train  going  at  the  rate  of  s  feet  per  second  round 
a  curve  of  radius  r  feet  the  outer  rail  should  be  raised 
h  inches  above  the  inner,  where  h  =  8^G/SS6  r,  in  which  G  is 
the  number  of  feet  in  the  gage  of  the  track.  Taking  s  =  60, 
r=2700,  and  G  =  i.7,  find  the  value  of  7i. 

Find  the  value  of  each  of  the  folloiviny : 

^    3  X  50  ,    14  X  5  X  81  „    130  x  65  x  7 

3. r-.  5. 


4. 


5x6  *     5  X  7  X  9  *      300  X  24 

2x7  ^    4  X  40  X  59  „     65  x  92.5 


3x11  '6x8x7  22.5  x  47.5 


162  THE  SLIDE  KULE 

Proportion.  One  of  the  features  of  the  slide  rule  is  that, 
however  the  slide  is  placed,  all  numbers  on  the  slide  are 
proportional  to  the  numbers  on  the  rule  that  coincide  with 
them.  That  is,  when  2  is  over  4,  3  is  over  6,  4  is  over  8, 
2.5  is  over  5,  1.2  is  over  2.4,  and  so  on;  in  other  words, 

2 :  4  =  3  :  6  =  4  :  8  =  2.5  :  5  =  1.2  :  2.4  =  . .  .. 

We  can  therefore  use  the  slide  rule  in  solving  proportions. 
For  example,  given  1.5:  3.7=  4.1:  a;,  find  the  value  of  x. 

We  set  1.5  over  3.7,  and  read  10.11  under  4.1.  In  this  case,  since  the 
result  exceeds  10,  the  limit  of  the  D  scale,  we  use  the  A  and  B  scales. 

We  may  express  the  rule  diagrammatically  as  follows : 


c 

Set  first  term 

Then  under  third  term 

D 

Over  second  term 

Read  fourth  term 

Exercises.    Proportion 

1.  Given  that  1  kg.  is  equivalent  to  2.2  lb.,  set  the  slide 
so  as  to  read  the  pounds  corresponding  to  kilograms. 

2.  A  certain  map'is  drawn  to  the  scale  1"=  80  mi.   Set  the 
slide  so  as  to  read  the  miles  corresponding  to  inches. 

3.  Given  that  1  m.  is  equivalent  to  39.37",  set  the  slide 
so  as  to  read  the  inches  corresponding  to  meters. 

4.  If  the  wages  of  7  men  for  1  da.  are  |31.50,  find  the 
wages  of  25  men  for  1  da.  at  the  same  rate. 

5.  If  a  factory  can  turn  out  6480  pairs  of  shoes  in  6  da., 
how  long  will  it  take  to  fill  an  order  for  36,000  pairs? 

Find  the  value  of  x  in  each  of  the  following  proportions  : 

6.  7:3  =  5:  ir.  8.  a;:  9  =17:  6.3.        10.  0.5:7  =  3:  2^. 

7.  2::  7=  2.3:4.        9.  5:8=7.2:rr.  11.  a::  0.7=  2.3  :  4, 


SQUARES  AND  SQUARE  ROOTS 


163 


Squares  and  Square  Roots.  By  examining  the  slide  rule 
we  see  that  the  numbers  on  A  are  the  squares  of  the  num- 
bers just  below  them  on  D.  Hence,  to  find  the  square  of  any 
number  we  place  the  hair  line  of  the  runner  over  the  num- 
ber on  D  and  read  the  square  under  the  hair  line  on  A. 

Conversely,  to  find  the  square  root  of  any  number  we  place 
the  hair  line  of  the  runner  over  the  number  on  A  and  read 
the  square  root  under  the  hair  line  on  D. 

The  slide  rule  may  also  be  conveniently  used  for  the  pur- 
pose of  evaluating  such  expressions  as  a%.  For  example,  to 
find  the  value  of  8^  x  5  we  have  the  following  arrangement : 


A 

Read  320.  Ans. 

B 

Set  1  (right) 

Over  5 

C 

D 

Over  8 

That  is,  we  set  1  on  ^  over  8  on  D,  and  over  5  on  ^  read  the  answer, 
320,  on  ^.   In  this  way  we  can  find  the  value  of  irr^,  the  area  of  a  circle. 

We  can  also  find  the  value  of  an  expression  in  the  form 
of  Vajh.   Thus,  to  find  the  value  of  V|  we  have  the  following : 


A 

Under  3  . 

B 

Set  4 

Under  1  (right) 

C 

D 

Read  0.866.  Ans. 

That  is,  we  set  ^  on  B  under  ^  on.  A,  and  under  \  on  B  read  the 
answer,  0.866,  on  D.  In  this  way  we  can  easily  find  the  value  of  Vo/tt  ; 
that  is,  given  the  area  of  a  circle  we  can  find  the  radius. 


164  THE  SLIDE  RULE 

Exercises.   Review 
Using  the  slide  rule,  find  the  square  of  each  of  the  following  : 

1.  4.  3.  7.  5.  16.7.  7.  38.5.  9.  61.7. 

2.  8.  4.  12.  6.  17.8.  8.  5.29.  10.  3.14. 

Find  the  square  root  of  each  of  the  folloiving : 

11.  4.  13.  16.  15.  64.  17.  144.  19.  14.4. 

12.  9.  14.  49.  16.  81.  18.  1.44.         20.  3.81. 

21.  Find  the  value  of  ttt^  when  r  =  28.2''. 

22.  Fmd  the  area  of  a  circle  which  has  a  radius  of  2.7". 

23.  Find  the  radius  that  should  be  used  in  drawing  a  circle 
which  shall  have  an  area  of  29  sq.  in. ;   an  area  of  42  sq.  in. 

Find  the  value  of  each  of  the  following : 

24.  3.1  X  16.12.  26.  32  x  6.5^.       '       28.  9  x  2.752. 

25.  3.34  X  2.92.  27.   7.2  x  232.  29.  0.7  x  2.22. 

30.  Find  the  diameter  of  a  cylindric  iron  rod  which  has 
a  cross-section  area  of  9.3  sq.  in. 

31.  What  is  the  cross-section  area  of  a  cylindric  iron  rod 
which  has  a  diameter  of  4.7"?  a  radius  of  1.5"? 

Using  the  slide  rule,  find  the  value  of  each  of  the  following : 

32.  7sin41^         35.  15  sin  45°.  38.  12  sin  17°  24'. 

33.  9  tan  25°.        36.  16  tan  50°.  39.  2.6  cos  36°  20'. 

34.  8  cos  27°.        37.  2.8  cos  35°.         40.  32.8  tan  58°  5'. 

To  find  7  sin  41°,  first  find  sin  41°  =  0.6561  on  page  144,  and  then 
multiply  0.6561  by  7.  It  will  thus  be  seen  that  the  slide  rule  is  often 
convenient  when  using  trigonometry. 


41.  Find  the  quotient  of  2V576  divided  by  sin  45°. 

42.  Find  the  value  of  }  cos  35°  ;  of  0.2  ^  sin  20°. 


CHAPTER  VI 

GENERAL  APPLICATIONS 
Exercises.   Review 

1.  There  are  48  rivets  in  a  length  of  17'  8|''  measured 
along  a  riveted  seam.    Find  the  pitch  of  the  rivets. 

The  pitch  is  the  distance  apart,  measured  between  centers. 

2.  From  a  large  can  containing  4|  gal.  of  oil  the  toolroom 
boy  fills  five  cans  which  hold  |  gal.  each,  and  one  which  holds 
1^ gal.    How  much  oil  is  left  in  the  large  can? 

3.  How  many  linear  feet  of  molding  will  be  needed  for  a 
room  23'  9''  X  31'  8''  ?  for  a  room  32'  6"  x  38'  9"  ? 

4.  Find  the  cost  of  2270'  of  pine  lumber  at  $105  per  M. 

5.  At  6|0  per  pound,  how  many  pounds  of  iron  can  be 
bought  for  1175  ?  for  |350  ?  for  |475  ? 

Results  of  this  kind  should  be  given  only  to  the  nearest  pound. 

6.  An  electrician's  helper  works  7|  hr.  a  day  at  66|^  an 
hour.    How  much  does  he  receive  per  day  ? 

7.  If  375  lb.  of  fire  clay  cost  $2.50,  how  much  will  1000  lb. 
cost  ?  How  much  will  2|-  T.  cost  ? 

Results  of  this  kind  should  be  given  only  to  the  nearest  cent. 

8.  How  many  planks  7"  wide  can  be  laid  side  by  side 
across  a  beam  16' 11"  long? 

9.  If  it  takes  \\  lb.  of  Babbitt  metal  for  one  bearing,  how 
many  pounds  will  it  take  to  babbitt  920  bearings  ? 

165 


166 


GENERAL  APPLICATIONS 


10.  A  stream  is  164'  wide  and  averages  6|'  in  depth.  If  it 
has  a  flow  of  3|-  mi./hr.,  how  many  gallons  of  water  flow 
by  in  1  hr.  ?    How  many  tons  of  water  flow  by  in  24  hr.  ? 

Water  weighs  62 J  Ib./cu.  ft.,  and  1  cu.  ft.  contains  7Jgal. 

11.  At   11^  a  pound,   find   the  cost  of       ^^.^ ~^,^^^ 

150  blank  nuts  made  of  cold-rolled  steel  f^^^^---...^,^.^-^^! 

weighing  0. 2816  Ib./cu.  in.,  the  blanks  being  ^--^^  |   ^,^^ 
3|''  square  and  1|''  thick,  and  the  center 

hole,  as  shown  in  the  figure  above,  being  2i^''  in  diameter. 

12.  Find  the  weight  of  a  cast- 
iron  pulley  of  the  dimensions  here 
given.  At  6|^  a  pound,  find  the 
cost  of  1675  pulleys  of  this  type. 

The  student  should  use  0.26  lb.  as 
the  weight  of  1  cu.  in.  of  cast  iron. 

13.  An  automobile  manufacturer  makes  115  lb.  of  steel 
into  ball  bearings  |"  in  diameter.    How  many  does  he  make  ? 

Take  0.2816  lb.  as  the  weight  of  1  cu.  in.  of  this  kind  of  steel. 

14.  At  500  cu.  ft.  to  the  ton,  how  many  tons  of  hay  will 
fill  a  hay  mow  27'  9"  long,  19'  3"  wide,  and  11'  6"  high  ? 

15.  At  7J-(^  a  pound,  what  is 
the  cost  of  75  steel  lathe  face- 
plates of  the  kind  here  shown  ? 

Use  the  weight  of  steel  given  in 
Ex.  13  above. 


16.  In  Ex.  15  suppose  that  the 
price  were  81  (^.  a  pound. 

17.  If  a  mechanical  trench  digger  removes  earth  at  the  rate 
of  1450  cu.  yd./hr.,  how  many  days  of  8  hr.  each  will  it  take 
to  dig  a  trench  61'  wide,  7'  deep,  and  81  mi.  long? 


REVIEW  EXERCISES  167 

18.  If  brass  castings  shrink  -^^'^  per  foot  of  finished  dimen- 
sions, find  the  length  of  the  pattern  for  each  of  the  following 
lengths  of  finished  castings:  2^  9^  9' 4^  5' 3^  1' 11^ 

19.  Find  the  length  of  the  piece  of  stock  necessary  to 
make  25  machine  bolts,  each  1^^  long,  allowing  |^"  on  each 
bolt  for  cutting  and  finishing. 

20.  If  a  motor  makes  560  R.P.M.,  how  many  revolutions 
does  it  make  in  8  hr.  ? 

21.  If  carriage  bolts  of  a  certain  size  weigh  19.2  lb.  per 
100,  what  is  the  weight  of  1975  bolts  of  this  size?  How 
many  bolts  will  it  take  to  weigh  384  lb.  ? 

22.  Neglecting  any  allowance  for  the  seams,  find  the  cost 
of  the  sheet  iron  required  for  a  smokestack  25^  high  and  20" 
in  diameter  at  29^  per  square  foot. 

23.  On  the  arbor  of  a  milling  machine  a  machinist  places 
five  collars  measuring  respectively  0.434'^,  0.968'^,  0.250", 
0.625",  0.5156".    Find  the  total  length  of  the  collars. 

24.  How  many  pieces,  each  1.45"  long  and  the  width  of 
the  plank,  can  be  sawed  from  a  plank  16.75'  long,  making  no 
allowance  for  waste  ?    How  long  a  piece  is  left  over  ? 

25.  On  a  lathe  making  175  R.P.M.  the  cutting  tool 
advances  0.015"  per  revolution.  How  far  does  the  tool  travel 
in  1  min.  ?   in  8  min.  ?   • 

26.  If  a  contractor  receives  $973.08  for  excavating  a 
cellar  at  $1.06  per  cubic  yard,  how  many  cubic  yards  does 
he  excavate  ? 

27.  In  Ex.  26  suppose  that  the  cellar  were  twice  as  long, 
twice  as  wide,  and  50%  deeper,  how  much  would  the  con- 
tractor receive  at  the  same  rate  ? 

28.  In  Ex.  27,  if  the  rate  were  $1.1 2|-  per  cubic  yard,  how 
much  would  the  contractor  receive  ? 


168  GENEKAL  APPLICATIONS 

29.  A  casting  weighed  625^  lb.  before  it  was  machined, 
and  598|  lb.  when  finished.  If  it  cost  37|^  per  pound  to 
produce  the  finished  casting,  including  labor  but  not  count- 
ing the  value  of  the  scrap,  and  the  scrap  was  sold  at  3|(f 
per  pound,  what  was  the  actual  cost  of  production  after  the 
value  of  the  scrap  was  deducted  ? 

30.  How  many  tons  of  iron  conduit  weighing  1066  Ib./lOO' 
are  needed  in  a  building  for  wliicli  tlie  specifications  require 
6270'  of  the  conduit? 

31.  How  many  pieces,  each  3.1 5^'  long,  can  be  sawed  from 
a  steel  bar  27.9'  long,  the  thickness  of  the  saw  being  0.162'', 
and  how  long  a  piece  will  be  left  over  ? 

32.  How  many  shingles  does  a  carpenter  use  in  shingling 
a  barn  if  he  lays  88  rows  with  128  shingles  to  a  row?  If 
when  buying  the  shingles  he  gave  his  order  to  the  next  higher 
iM  required,  find  the  cost  of  the  shingles  at  |9.87  per  M. 

33.  The  width  of  a  street  was  54.6'  after  improvements 
were  made  by  which  the  street  was  widened  5%.  How  wide 
was  the  street  before  the  improvements  were  made? 

34.  If  soft-steel  bars  are  selling  at  $3.57  per  100  lb.,  and 
if  they  sold  a  year  ago  at  $4.20,  wliat  is  the  per  cent  of 
decrease  in  price  ? 

35.  Find  the  cost  of  laying  a  cement  walk  4'  11"  wide 
and  50' 8"  long,  at  331^  per  square  foot;  at  37|^  per 
square  foot;   at  37|(f  per  square  foot. 

36.  A  builder  contracted  to  erect  a  house  for  $9450. 
When  the  house  was  completed  he  found  that  the  actual 
cost  was  $8032.50.  Find  the  rate  of  profit  on  the  contract 
price;  on  the  cost  price. 

37.  How  many  feet  of  lumber  will  it  take  to  make  3  doz. 
drawing  boards,  each  2'  3"  x  3'  2"  x  |"  ? 


REVIEW  EXERCISES  169 

38.  How  many  revolutions  will  an  automobile  wheel  32" 
in  diameter  make  while  the  car  is  traveling  1  mi.? 

In  such  cases  the  slipping  of  the  wheel  and  similar  factors  are  not  to 
be  considered  unless  the  contrary  is  expressly  stated. 

39.  If  a  compositor  earns  |207  in  4  wk.,  how  many  weeks 
will  it  take  him  to  earn  |1242  ? 

40.  The  stock  for  a  certain  job  of  printing  cost  $27.90, 
and  the  printing  itself  cost  $11.25.  If  the  printer  figures 
18%  profit  on  the  stock  and  35%  profit  on  the  printing, 
these  items  of  profit  to  cover  his  profit  and  all  overhead 
and  other  charges,  for  what  amount  does  he  bill  the  job  ? 

Such  a  bill  would,  in  ordinary  practice,  be  figured  to  the  next  higher 
25^  or  50^,  although  on  a  larger  order  the  bill  might  be  figured  to  the 
next  higher  |1.    In  this  case  take  the  next  higher  25^. 

41.  Making  no  allowance  for  waste,  how  many  sheets  of 
blueprint  paper,  each  11"  x  15",  can  be  cut  from  a  roll  30' 
wide  and  10  yd.  long  ? 

42.  The  diameters  of  a  #10  wire  are  given  on  different 
wire  gages  as  follows: 

Brown  &  Sharpe 0.1019" 

Birmingham 0.1340" 

Washburn 0.1350" 

Trenton 0.1300" 

Imperial 0.1280" 

Which  two  gages  differ  the  most  in  size,  and  by  how  much 
do  they  differ  ? 

43.  If  a  dealer  gains  20%  on  the  cost  of  a  battery  when 
he  sells  it  for  60^,  what  per  cent  would  he  gain  on  the  cost 
if  he  should  sell  it  for  65 (^? 

44.  If  a  train  travels  369  mi.  in  9  hr.,  how  far  does  it  travel 
at  the  same  rate  in  23  hr.  15  min.? 


170  GENERAL  APPLICATIONS 

45.  In  a  lot  of  950  cast-iron  pulleys  7^%  are  rejected  on 
account  of  defects.  How  many  pulleys  are  rejected  and  how 
many  are  accepted? 

Since  7^%  of  950  gives  a  number  involving  a  decimal,  the  student 
should  use  his  common  sense  and  take  the  nearest  whole  number.  The 
per  cent  rejected  cannot  then  be  exactly  7  J,  but  in  practical  work  we 
should  rarely  say  that  7^^%  of  the  pulleys  were  rejected. 

46.  At  9  Ib./ft,  find  the  total  weight  of  6  pieces  of  pipe 
3'  6"  long,  7  pieces  4'  2"  long,  and  5  pieces  6'  9'^  long. 

47.  If  a  dealer  buys  zinc  sheets  at  $215.50  per  ton,  at 
what  price  per  pound  must  he  sell  them  in  order  to  gain 

45%  on  the  cost? 

48.  How  long  will  it  take  a  pump  delivering  1.6  gal.  of 
water  at  a  stroke  and  making  75  strokes  per  minute  to  pump 
1500  gal.  of  water? 

49.  An  electrical-appliance  dealer  bought  the  following 
bill  of  goods:  45  cast-bronze  push  buttons  @  56^  each; 
5625'  #18  annunciator  wire,  15071b.,  @  48(f  a  pound;  30 
steel  outlet  boxes  @  |21.65  per  100.  Make  out  the  bill  and 
find  the  total  cost  of  the  goods. 

50.  If  in  Ex.  49  the  dealer  took  advantage  of  a  2i%  dis- 
count for  cash  within  10  da.,  how  much  did  he  save  ?  What 
was  the  net  amount  which  he  paid  for  the  goods  ? 

51.  Allowing  an  average  space  8'xl2'  for  each  car,  this 
allowance  covering  the  aisle  space,  how  many  cars  can  be 
stored  in  a  garage  90'  square  ? 

52.  A  carpenter's  wages  were  increased  15%,  the  increase 
amounting  to  75^  a  day.  How  much  was  he  getting  per  day 
before  the  increase  and  how  much  was  he  getting  thereafter  ? 
If  his  wages  were  later  decreased  8%,  how  much  was  he  then, 
getting  per  day  ? 


REVIEW  EXERCISES  171 

53.  If  lead  is  worth  $98  per  ton,  how  much  is  750  lb. 
worth?  If  a  dealer  bought  lead  at  this  rate  and  sold  it  at 
8  ^  a  pound,  what  per  cent  of  profit  did  he  make  on  the  cost  ? 

As  already  stated,  a  ton  is  to  be  taken  as  2000  lb.  unless  the  long  ton 
of  2240  lb.  is  specifically  mentioned. 

54.  Find  the  weight  in  tons  of  the  rails  required  for  1  mi. 
of  double-track  railway,  the  rails  weighmg  120  lb.  per  yard. 

55.  If  the  speed  of  an  engine  which  is  running  at  the  rate 
of  98  R.P.M.  is  increased  8i%,  what  is  the  number  of  R.P.M. 
after  the  speed  is  increased  ? 

In  problems  dealing  with  the  speeds  of  engines,  pulleys,  and  the  like, 
the  result  should  be  given  to  the  nearest  whole  number. 

56.  If  the  speed  of  an  engine  which  is  running  at  the  rate 
of  400  R.P.M.  is  increased  10%,  what  is  then  the  number 
of  R.P.M.?  If  this  new  speed  is  decreased  10%,  what  is 
then  the  number  of  R.P.M.? 

57.  By  selling  a  lathe  for  $183  above  cost  a  dealer  gained 
15%  on  the  cost.    How  much  did  the  lathe  cost  the  dealer? 

58.  By  selling  a  lathe  for  $1840  a  dealer  gained  15%  on 
the  cost.    How  much  did  the  lathe  cost  the  dealer? 

59.  An  agent  bought  three  motors  for  |120,  $160,  and 
$190  respectively.  He  sold  the  first  at  a  loss  of  8%,  the 
second  at  cost,  and  the  third  at  a  profit  of  6%.  Find  his 
total  profit  or  loss  on  the  motors. 

60.  A  contractor  bought  75  M  bricks  at  $11.50  per  M. 
If  he  sold  I  of  the  bricks  at  |  their  cost,  and  sold  the  rest 
for  $500,  how  much  did  he  lose  ? 

61.  If  a  piece  of  iron  8'  long,  6''  wide,  and  4''  thick  weighs 
600  lb.,  what  is  the  weight  of  a  piece  of  iron  that  is  13'  long, 
8''wide,  and  6^' thick? 


172  GENERAL  APPLICATIONS 

62.  Find  the  cross-section  area  of  the  cast-iron  beam  shown 
in  the  blueprint  on  page  173. 

In  such  cases  neglect  entirely  the  rounding  of  the  corners. 

63.  How  many  square  feet  are  there  in  the  floor  of  the 
machine  shop,  the  plan  of  which  is  shown  in  the  blueprint? 

64.  If  there  are  15  machines  in  the  shop  in  Ex.  63,  what  is 
the  average  number  of  square  feet  of  floor  space  per  machine? 

65.  In  the  machine  shop  in  Ex.  63  it  is  desired  to  install 
six  new  machines,  each  of  which  requires  275  sq.  ft.  of  floor 
space.  If  the  additional  floor  space  is  obtained  by  building 
an  addition  along  the  entire  length  of  the  right  side  of  the 
shop,  how  wide  is  the  addition  ? 

66.  Find  the  area  of  the  boiler  patch  shown  in  the  blueprint. 

67.  If  the  boiler  has  a  pressure  of  225  Ib./sq.  in.,  and  the 
boiler  patch  covers  an  opening  8J"  square,  how  much  press- 
ure is  exerted  against  it  ? 

68.  The  outside  row  of  rivets  on  the  boiler  patch  is  to 
extend  all  the  way  around  the  patch  as  indicated  in  the 
blueprint.  How  many  rivets  are  needed  for  this  row  if  the 
rivets  are  to  be  spaced  ly  apart? 

69.  The  gusset  plate  shown  in  the  blueprint  is  made  of 
#4  U.S.  standard-gage  steel  weighing  9.525  lb. /sq.  ft.  Find 
the  weight  of  the  plate. 

70.  If  each  side  of  the  gusset  plate  were  half  as  long  again, 
what  would  then  be  the  weight  of  the  plate  ? 

71.  The  outlet  box  of  a  heating  system  is  rectangular  in 
shape  and  is  4'  2^^  long  and  14'^  wide.  A  pipe  with  a  square 
cross  section  leads  from  this  box,  and  the  area  of  the  cross 
section  is  the  same  as  the  area  of  the  bottom  of  the  outlet 
box.    Find  the  length  of  a  side  of  the  square  pipe. 


REVIEW  EXERCISES 


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174  GENERAL  APPLICATIONS 

72.  If  3L''  is  turned  off  the  outside  of  the  piston  ring 
shown  in  the  blueprint  on  page  175,  by  how  many  square 
inches  is  the  area  of  the  cross  section  of  the  ring  reduced  ? 

73.  Find  the  cross-section  area  of  the  metal  in  the  steam 
pipe  shown  in  the  blueprint. 

74.  Find  the  internal  cross-section  area  of  the  steam  pipe 
shown  in  the  blueprint. 

75.  Find  the  number  of  square  feet  in  the  head  of  the 
boiler  shown  in  the  blueprint,  without  deducting  the  area 
taken  up  loj  any  of  the  openings. 

76.  The  inside  diameter  of  each  tube  being  2.265'^,  find 
the  total  area  in  square  feet  of  the  internal  cross  sections  of 
the  tubes.    Find  the  area  of  the  opening  which  is  16''  x  22''. 

77.  A  cast-iron  pulley  weighing  0.26  Ib./cu.  in.  has  a  cored 
center  hole  3| "  in  diameter  and  9"  long.  By  how  many  pounds 
does  this  hole  reduce  the  weight  of  the  casting  ? 

78.  When  the  steam  pressure  is  95  Ib./sq.  in.,  Avhat  is  the 
total  pressure  exerted  on  a  7|^-inch  piston  ? 

This  means  that  the  diameter  of  the  piston  is  7  J/'. 

79.  A  weight  is  supported  by  three  rods,  each  of  which 
has  a  diameter  of  2|".  Assuming  that  a  single  rod,  with  a 
cross-section  area  equal 'to  the  combined  cross-section  areas 
of  the  three  rods,  would  support  the  same  weight,  what  should 
be  the  diameter  of  that  rod  ? 

80.  From  a  rectangular  piece  of  sheet  iron  8'  3"  long  and 
3'  10"  wide  eight  disks  are  cut,  each  disk  being  22"  in 
diameter.  Making  no  allowance  for  waste,  find  the  number 
of  square  inches  of  sheet  iron  left  in  the  original  piece. 

81.  A  flywheel  made  of  iron  weighing  450  lb. /cu.  ft.  has 
an  outside  diameter  of  15'.  The  rim  is  12"  wide  and  has  a 
radial  thickness  of  7".    Find  the  weight  of  the  rim. 


REVIEW  EXERCISES 


175 


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HEAD  of    BOILEf=\ 


176  GENERAL  APPLICATIONS 

Exercises.    Automobile  Transmission 

1.  The  blueprint  on  page  177  shows  part  of  the  gear  con- 
nections of  a  4-speed  selective  automobile  transmission.  The 
power  is  transmitted  from  the  engine  through  the  sleeve  A 
to  the  gear  B  and  thence  to  the  countershaft  H  by  the  gear  C. 
The  engagement  of  the  gears  is  explained  below.  If  ^  makes 
1000  R.P.M.,  what  is  the  speed  of  the  countershaft  H? 

In  the  blueprint  the  gear  connections  of  three  of  the  forward  speeds 
are  shown.  The  main  shaft  (7  is  a  square  shaft  which  runs  independently 
of  sleeve  A,  and  the  gears  on  this  shaft  are  so  arranged  that  they  can 
be  moved  into  position  to  engage  the  corresponding  gears  oti  the  counter- 
shaft. Thus,  D  (64  T)  can  be  brought  into  engagement  with  D  (32  T) 
for  low  speed.  The  position  of  the  gears  shown  in  the  blueprint  is 
called  *'  neutral "  since  no  gear  on  the  main  shaft  is  engaged,  and  the 
countershaft  only  is  being  revolved  by  the  engine.  For  high  speed 
there  are  projections  called  "  dogs  "  on  the  front  of  gear  F  on  the  main 
shaft  and  on  the  back  of  gear  B  which  can  be  brought  into  engage- 
ment, thus  driving  the  main  shaft  direct  from  the  engine.  For  reverse 
an  intermediate  gear  is  thrown  up  from  below  between  gears  D,  D,  thus 
reversing  the  direction  of  rotation  of  the  main  shaft.  The  horizontal 
shaft  driven  by  the  bevel  gears  is  connected  through  the  differential 
(not  shown)  to  the  rear  wheels. 

2.  In  Ex.  1  what  is  the  speed  of  the  main  shaft  G  when 
the  low-speed  gears  D,  Z>  are  engaged?  when  the  second- 
speed  gears  E,  E  are  engaged  ?  when  the  third-speed  gears 
F^  F  are  engaged  ? 

3.  Find  the  speed  of  the  countershaft  when  the  engine 
is  running  at  750  R.P.M. 

4.  In  Ex.  3  find  the  speed  of  the  main  shaft  for  low  speed; 
for  second  speed ;  for  third  speed. 

5.  Consider  Ex.  4  when  the  engine  runs  at  1400  R.P.M. 

6.  In  Ex.  5  what  is  the  speed  of  the  main  shaft  when  the 
intermediate  reverse  gear  is  engaged  between  the  gears  D^I)'^. 
when  the  dogs  on  gears  B  and  F  are  engaged  for  high  speed  ? 


AUTOMOBILE  TRANSMISSION 


177 


1T8  GENERAL  APPLICATIONS 

Exercises.   Apartment-House  Structure 

1.  In  excavating  the  cellar  for  the  apartment  house,  the 
floor  plan  of  which  is  shown  on  page  179,  the  excavation  was 
carried  1'  beyond  each  outside  wall  and  8'  below  the  surface 
of  the  lot.    Find  the  number  of  loads  of  earth  removed. 

Consider  a  load  as  1  cu.  yd.  In  making  the  excavation  the  space 
occupied  by  the  two  fire  escapes  in  the  court  is  not  considered. 

2.  For  the  cellar  floor  the  contractor  used  a  layer  of  con- 
crete 1^"  thick  extending  to  the  inside  of  the  outer  founda- 
tion walls,  which  are  18'^  thick.  Find  the  number  of  cubic 
feet  of  concrete  used  in  laying  the  cellar  floor. 

3.  Find  the  number  of  feet  of  |-inch  lumber  required  to 
floor  all  the  bedrooms  on  this  floor  of  the  apartment  house, 
allowing  i  extra  for  waste. 

4.  The  kitchen  floors  are  to  be  covered  with  linoleum  cost- 
ing 37J  (f  per  square  foot.  Find  the  total  cost  of  the  linoleum 
for  the  three  kitchens  on  this  floor. 

In  this  problem  first  calculate  the  amount  of  linoleum  required  for 
each  kitchen,  disregarding  any  allowance  for  dressers,  cupboards,  and 
the  like,  to  offset  the  waste  in  cutting  and  laying.  If  these  results  con- 
tain fractions,  use  the  next  larger  whole  numbers. 

5.  If  the  rooms  are  9^  high,  find  the  total  number  of  square 
yards  of  plastering  required  for  all  the  livmg  rooms  and 
bedrooms,  making  no  allowance  for  doors  and  windows. 

6.  The  dining  rooms  are  to  have  oak  flooring.  Allowing 
1-  extra  for  waste,  find  the  number  of  feet  of  |-inch  flooring 
required  for  all  three  dining  rooms.  Fmd  the  cost  of  this 
flooring  at  |135  per  M. 

7.  Each  dining  room  is  to  have  a  plate  rail  around  the 
walls.  How  many  running  feet  will  be  needed  for  the  three 
dining  rooms,  making  an  allowance  of  10%  of  the  length  for 
each  room  for  doors  and  windows  ? 


APARTMENT-HOUSE  STRUCTURE  179 


180  GENERAL  APPLICATIONS 

Exercises.    Pulley  and  Gear  Trains 

1.  In  the  pulley  train  shown  in  the  blueprint  on  page  181 
find  the  number  of  R.P.M.  of  the  emery  wheel. 

2.  In  the  same  pulley  train  find  the  number  of  R.P.M. 
of  the  hack  saw. 

3.  If  a  new  motor  with  a  speed  of  650  R.P.M.,  but  having 
the  same  size  of  driving  pulley  as  the  old  motor,  were  installed 
to  drive  the  pulley  train,  what  would  then  be  the  speed  of 
the  emery  wheel  ? 

4.  With  the  new  motor  of  Ex.  3  what  would  be  the 
number  of  R.  P.  M.  of  the  hack  saw  ? 

5.  If  it  is  desired  to  increase  the  speed  of  the  hack  saw 
in  Ex.  4  by  25  R.  P.  M.,  what  size  pulley  should  replace  the 
18-inch  pulley  on  the  line  shaft? 

6.  In  Ex.  3  what  size  pulley  to  the  nearest  I"  should 
replace  the  24-inch  pulley  on  the  line  shaft  in  order  to  have 
the  speed  of  the  emery  wheel  the  same  as  in  Ex.  1  ? 

7.  Find  the  number  of  R.P.M.  of  the  12-inch  pulley  in 
the  blueprint  of  the  pulley  and  gear  train. 

8.  If  the  speed  of  the  line  shaft  in  Ex.  7  were  reduced 
to  150  R.P.M.,  what  would  then  be  the  speed  of  the  12-inch 
pulley?    of  the  48-Tgear? 

9.  If  the  pulley  on  the  spindle  of  a  lathe  is  4^  in  diam- 
eter, and  the  countershaft  carrying  a  pulley  lOf"  in  diameter 
makes  195  R.P.M.,  what  is  the  speed  of  the  lathe  spindle? 

10.  A  stepped-cone  pulley  on  the  driving  spindle  of  a 
lathe  has  diameters  of  4l-^^  6f'',  7^',  and  9|'\  and  is  belted 
to  a  similar  pulley,  the  diameters  of  which  are  the  same  but 
in  reverse  order,  on  the  countershaft.  If  the  countershaft 
makes  165  R.P.M.,  what  is  the  number  of  R.P.M.  of  the 
spindle  at  each  of  the  four  different  speeds? 


PULLEY  AND  GEAR  TRAINS 


181 


182 


GENEEAL  APPLICATIONS 


Exercises.    Conduit  Wiring 

1.  From  the  data  in  the  blueprint  on  page  183  insert  the 
quantities  required  in  the  following  stock  bill  for  wiring : 


Quantity 

Description 

Price 

|-inch  conduit 

117.50  per  100' 

1-inch  conduit 

$12.50  per  100' 

|-inch  condulets,  type  A 

149  per  100, 
less  21% 

J-inch  condulets,  type  A 

$46  per  100, 

less  21% 

1-inch  condulets,  type  G 

164.50  per  100, 
less  2^% 

i-inch  condulets,  type  L 

$55  per  100, 
less  2^% 

|-inch  outlet  boxes 

34  (^  each, 

less  6% 

Pilot  light,  type  J 

$115  per  100, 
less  21% 

Plug-fuse  cut-out 

$72  per  100 

6-circuit  panel  board 

$15.20 

Incandescent-lamp  cord 

$71.30  per  1000' 

/14  R.C.  wire,  2  wires  in 

all  |-inch  conduit 

$22.40  per  1000' 

/lO  R.C.  wire,  3  wires  in 

all  |-inch  conduit 

$39.30  per  1000' 

In  figuring  the  length  of  the  conduit  and  wire  needed,  compute  as  a 
whole  foot  any  fraction  of  a  foot  in  the  total  length  required. 

2.  Find  the  total  cost  of  all  the  items  in  the  above  bill. 


CONDUIT  WIEING 


183 


Outlet  Box 


5'-f    'f 


Lamp  Cord 


Ti^fje  A  Panel  Board 

:§^:5:    .r  BASEMENT 


184 


GENERAL  APPLICATIONS 


Exercises.    Boiler  Connections 

1.  From  the  data  shown  m  the  blueprint  on  page  185  insert 
the  different  quantities  required  in  the  following  stock  bill : 


Quantity 

Description 

Price 

^-inch  galvanized  pipe 

110  per  foot 

|-inch  galvanized  pipe 

130  per  foot 

, 

1-inch  galvanized  pipe 

190  per  foot 

|-inch  galvanized  elbows 

$1.25  per  dozen 

|-inch  45° galvanized  elbows 

11.30  perdozen 

1 X  |-inch  galvanized  elbows 

$1.70  per  dozen 

1-inch  galvanized  unions 

$42  per  100 

lx|-inch  galvanized  unions 

$35  per  100 

|-inch  unions 

$30  per  100 

|-inch  tees 

$1.30  perdozen 

|x^x|-inch  tees 

$1.30  per  dozen 

^-inch  couplings 

$1.10  perdozen 

|-inch  sediment  cock 

900  each 

|-mch  valve 

$1.75  each 

Boiler 

$19.20  each 

Stand 

$1.25  each 

In  figuring  the  length  of  pipe  needed,  compute  as  the  next  half  foot 
any  fraction  of  a  foot  in  the  total  length  required. 

2.  Find  the  total  cost  of  all  the  items  in  the  above  bill. 

3.  Approximately,  how  many  gallons  does  the  boiler  hold  ? 

Consider  the  boiler  as  a  cylinder  5'  in  height  and  disregard  the  fact 
that  the  top  is  rounded. 

4.  Find  the  approximate  weight  of  the  empty  boiler  which 
is  made  of  /7  gage  sheet  iron  weighing  7^  Ib./sq.  ft. 

Disregard  the  overlapping  seams  and  the  weight  of  the  rivets. 


BOILER  CONNECTIONS 


185 


XBoiler  Unic 


5'-0"\       L^M 


S!  ^Circulation  Fib 

'  '  /  or.  ' 

^T      J'   r2''4l^nion 


I  Union 

^1 

o  o  oo 

Ix^a/er  SacA" 


T^ediinent  V\ 
^Coujolinj^ 


Sediment  Pipe 


-4  ^f^  J  ice 
Stand 


BOILEFi  CONNECTIONS 


186 


GENERAL  APPLICATIONS 
Exercises.    Section  of  Lockers 


1.  From  the  data  given  in  the  blueprint  on  page  187  fill 
out  the  following  stock  bill  for  the  section  of  lockers : 


Number  of 
Pieces 

Length 

Width 

Thickness 

Description 

A,  top 

B,  door  stiles 

C,  door  top  rails 

D,  door  bottom  rails 

JS',  partitions 

F,  back 

G,  sides 

If,  upper  shelf 
7,  lower  shelf 

J,  frame 

K,  door  panel 

X,  molding 

M,  cleats 

JV,  upper  part  of  frame 

In  computing  the  amount  of  molding,  which  is  1|"  wide  and  1" 
thick,  figure  the  number  of  running  feet  required. 

2.  The  lockers  are  to  be  made  of  chestnut,  with  the  excep- 
tion of  the  back,  which  is  to  be  made  of  matched  whitewood 
strips.  The  chestnut  costs  tllO  per  M,  the  whitewood  $65 
per  M,  and  the  molding  10^  per  running  foot.  Find  the  total 
cost  of  the  lumber  for  the  section. 

3.  Allowing  1  gal.  of  paint  to  600  sq.  ft.  for  each  coat,  find 
the  amount  of  paint  required  for  the  section,  giving  one  coat 
inside  and  two  coats  outside. 


SECTION  OF  LOCKERS 


187 


188  GENERAL  APPLICATIONS 


Exercises.    Review 


1.  The  turret-lathe  cross  slide  shown  in  the  blueprint  on 
page  189  is  made  of  cast  iron  weighing  450  Ib./cu.  ft.  Find 
the  weight  of.  the  slide  before  the  slots  were  machined  in  it. 

2.  In  Ex.  1  find  the  weight  of  thie  cross  slide  after  all  the 
slots  are  machined. 

3.  The  dimensions  shown  in  the  blueprmt  of  the  shaper 
table,  which  is  made  of  cast  iron  weighing  450  lb./cu.  ft.,  rep- 
resent the  finished  size.  In  finishing  the  table  -^^"  was  first 
machined  ofP  each  outside  surface,  and  then  the  holes  and 
T-slots  were  cut.  Find  the  weight  of  the  casting  before  it 
was  machined. 

4.  The  six  holes  in  each  of  the  two  sides  were  drilled  and 
the  T-slots  were  milled  after  the  table  in  Ex.  3  was  machined. 
By  how  many  pounds  was  the  weight  of  the  table  reduced  in 
these  two  operations? 

5.  The  concrete  pipe-support  shown  in  the  blueprint  is  875' 
long.    Find  the  number  of  cubic  feet  of  concrete  required. 

6.  The  web  pulley  shown  in  the  blueprint  is  made  of  cast 
iron  weighing  450  lb./cu.  ft.  Find  the  cost  of  25  pulleys  of 
this  type  at  6|  (f  a  pound. 

7.  In  the  web  pulley  find  the  area  to  the  nearest  1  sq.  in. 
of  the  4-incli  face  of  the  rim  B. 

8.  Find  the  reduction  in  weight  of  the  web  pulley  if  eight 
l|-inch  holes  are  drilled  through  the  web  W. 

9.  Find  the  speed  in  F.  P.  M.  (feet  per  minute)  of  a  point 
on  the  rim  of  the  web  pulley  when  the  shaft  on  which  the 
pulley  is  fixed  is  making  85  R.  P.  M. 

10.  What  is  the  weight  of  a  pulley  made  of  steel  weighing 
490  lb./cu.  ft.,  if  each  dimension  of  the  steel  pulley  is  |  the 
length  of  the  corresponding  dimension  in  the  blueprint  ? 


REVIEW  EXERCISES 


189 


190  GENERAL  APPLICATIONS 

11.  The  flanged  shaft-coupling  shown  in  the  blueprint  on 
page  191  is  made  of  cast  iron  weighing  450  Ib./cu.  ft.,  and 
there  are  eight  of  the  -|-inch  holes  through  the  flange.  Find 
the  weight  of  25  couplings  of  this  type. 

12.  Find  the  area  of  each  of  the  two  1^-inch  faces  of  the 
cone  pulley  shown  in  the  blueprint. 

13.  In  the  cone  pulley,  which  is  made  of  cast  iron  weighing 
450  Ib./cu.  ft.,  the  web  A  is  |''  thick  and  is  pierced  by  six 
1-inch  holes.    Find  the  weight  of  the  web. 

14.  In  the  cone  pulley  the  cored  hub  B  is  3J"  long.  Find 
the  weight  of  the  hub. 

15.  Find  the  total  weight  of  the  cone  pulley. 

16.  The  dimensions  in  the  blueprint  of  the  flanged  pulley, 
which  is  made  of  cast  iron  weighing  450  Ib./cu.  ft.,  show  the 
size  of  the  pulley  as  cast.  In  finishing  the  pulley  I"  is  turned 
off  each  surface  except  the  |-inch  core.  Find  the  cost  of 
1250  finished  pulleys  at  7|^  a  pound. 

17.  Find  the  area  of  the  rim  B  of  the  flange  on  the  rough 
flanged  pulley  as  shown  in  the  blueprint.  Find  this  area  when 
the  pulley  has  been  finished  as  described  in  Ex.  16. 

18.  The  lathe  faceplate  is  made  of  cast  iron  weighing 
450  Ib./cu.  ft.,  and,  while  the  cored  slots  all  have  the  same 
width,  there  are  three  different  lengths,  as  shown  in  the 
blueprint.  Considering  all  the  slots  as  rectangular  openings, 
find,  approximately,  how  much  more  the  faceplate  would  have 
weighed  if  it  had  been  cast  solid. 

19.  Find  the  area  of  tlie  rim  of  the  lathe  faceplate. 

20.  Find  the  weight  of  metal  removed  in  boring  the  cored 
center  hole  in  the  lathe  faceplate  to  a  diameter  of  2^", 

21.  Find  the  speed  in  F.P.M.  of  a  point  on  the  rim  when 
the  faceplate  is  being  driven  at  40  R.P.M. 


BEVIEW  EXERCISES 


191 


192  GENERAL  APPLICATIONS 

22.  If  a  hack  saw  makes  96  strokes  per  minute,  how  many 
strokes  does  it  make  in  1|  hr.  ?  in  1  hr.  55min.? 

23.  The  driving  pulley  on  a  shaft  is  44"  in  diameter  and 
makes  240  R.P.M.  Find  the  speed  in  F.P.M.  of  a  point  on 
the  circumference  of  the  pulley.  Find  the  rim  speed  of  a 
12-inch  pulley  which  is  belted  to  the  large  pulley. 

24.  If  steel  expands  0.000007  of  its  length  for  each  degree 
Fahrenheit  that  it  increases  in  temperature,  computed  on  its 
length  when  it  begins  to  expand,  find  the  increase  in  the 
length  of  a  30-foot  steel  rail  Avhen  heated  from  10°  F.  to  120°  F. 

The  abbreviation  F.  means  Fahrenheit. 

25.  If  the  diameter  of  a  piston  is  28'^  and  the  pressure 
of  steam  in  the  cylinder  is  120  lb./sq.  in.,  what  is  the  total 
pressure  of  the  steam  upon  the  piston? 

26.  A  countershaft  has  upon  it  two  pulleys,  each  10"  in 
diameter,  and  the  speed  of  the  countershaft  is  500  R.P.M. 
Find  the  diameters  of  the  pulleys  of  two  machines  which, 
when  belted  to  the  two  pulleys  mentioned,  will  have  speeds 
of  200  R. P.M.  and  800  R.P.M.  respectively. 

27.  Experiment  has  shown  that  the  maximum  load  L  in 
tons,  which  can  safely  be  fastened  to  an  iron  chain  in  which 
the  diameter  of  the  chain  iron  is  d  inches,  is  expressed  by 
the  formula  L^^Ad!^,  Find  the  maximum  safe  load  that 
can  be  lifted  by  a  chain  in  which  d=0,75", 

28.  The  front  sprocket  wheel  of  a  certain  bicycle  has  26 
teeth  and  the  rear  sprocket  wheel  has  9  teeth.  If  the  rear 
tire  is  32"  in  diameter,  how  many  turns  of  the  pedals  will 
be  made  in  riding  the  bicycle  1  mi.  without  coasting  ? 

29.  A  pump  18"  in  diameter,  having  a  24-inch  stroke  and 
making  25  strokes  per  minute,  can  pump  how  many  cubic 
feet  of  water  per  hour  ?  how  many  gallons  ? 


TABLES  AND  RULES 

Length 
12  inches  (in.)  =  1  foot  (ft.) 
3  feet  =  1  yard  (yd.) 
5|  yards,  or  16|feet=  1  rod(rd.) 
320  rods,  or  5280  feet  =  1  mile  (mi.) 

Square  Measure 
144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 
9  square  feet  =  1  square  yard  (sq.  yd.) 
30|  square  yards  =  1  square  rod  (sq.  rd.) 
160  square  rods  =  1  acre  (A.) 

Cubic  Measure 
1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.yd.) 
128  cubic  feet  =  1  cord  (cd.) 

Weight 
16  ounces  (oz.)  =  1  pound  (lb.) 
2000  pounds  =  1  ton  (T.) 
2240  pounds  =  1  long  ton 

Liquid  Measure 
4  gills  (gi.)  =  1  pint  (pt.) 
2  pints  =  1  quart  (qt.) 
4  quarts  =  1  gallon  (gal.) 

Dry  Measure 
2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

193 


194  TABLES  AND  RULES 

Metric  Length 

1  kilometer  (km.)  =  1000  meters 

Meter  (m.) 
1  centimeter  (cm.)  =  0.01  meter 
1  millimeter  (mm.)  =  0.001  meter 

For  ordinary  comimrisons  we  usually  think  of  1  km.  as  0.6  mi. ;  of  1  m. 
as  39 ^'',  or  3 1';  of  1  cm.  as  0.4";  and  of  1  mm.  as  0.04''.  In  cases  requir- 
ing greater  accuracy  the  following  approximate  equivalents  may  be  used : 
1  km.  =  0.62  mi.,  1  m.  =  39.37",  1  cm.  =  0.394",  and  1  mm.  =  0.039". 

Metric  Weight 

1  metric  ton  (t.)  =  1000  kilograms 
1  kilogram  (kg.)  =  1000  grams 
Gram  (g.) 

The  following  approximate  equivalents  of  metric  weight  may  be 
used  :  1 1.  =  2204.6  lb.,  1  kg.  =  2.2  lb.,  and  1  g.  =  15.43  grains. 

Metric  Capacity 

1  hektoliter  (hi.)  =  100  liters 
Liter  (1.) 
1  centiliter  (cl.)  =  0.01  liter 

The  equivalent  of  1 1.  is  approximately  1  qt.  The  liter  is  the  volume 
of  a  cube  that  is  0.1  m.,  or  about  4",  on  an  edge. 

Angles  and  Arcs 

60  seconds  (60")  =  1  minute  (1') 
60  minutes  =  1  degree  (1°) 

Counting 

12  units  =  1  dozen  (doz.) 
12  dozen,  or  144  units  =  1  gross  (gr.) 
12  gross,  or  1728  units  =  1  great  gross 


TABLES  AND  RULES  195 

Common  Equivalents 
1  gal.  contains  231  cu.  in.,  or  0.134  cu.ft. 
1  cu.  ft.  contains  7|  gal. 
1  bbl.  contains  4|  cu.  ft.,  or  31 1  gal. 
1  bu.  contains  2150.42  cu.in.  (approximately  2150  cu.  in.). 
1  bu.  contains  approximately  l|cu.  ft. 
leu.  ft.  of  water  weighs  62.425  lb.  (approximately  62|lb.). 
1  gal.  of  water  weighs  8.345  lb.  (approximately  8|  lb.). 

A  ton  of  coal  varies  in  volume  according  to  kind  and  grade,  but  for 
general  purposes  the  volume  may  be  taken  as  35  cu.  ft. 

Convenient  Rules 
The  circumference  of  a  circle  is  ~  times  the  diameter. 
For  a  higher  degree  of  accuracy,  c  =  3.1416  d. 

The  diameter  of  a  circle  is  ~  of  the  circumferen.ce. 
For  a  higher  degree  of  accuracy,  d  —  0.3183  c. 

The  area  of  a  circle  is  y|  times  the  square  of  the  diameter. 
For  a  higher  degree  of  accuracy,  A  =  0.7854  d'^. 

The  height  of  an  equilateral  triangle  is  0.866  of  the  side. 
The  diagonal  of  a  square  is  1.414  times  the  side. 
In  the  shop  the  diagonal  of  a  square  is  also  called  "long  diameter" 
or  "  distance  across  the  corners." 

The  "  long  diameter  "  of  a  regidar  hexagon  is  tioice  the  side. 

The  '^  short  diameter ^^  or  the  perpendicidar  distance  hetiveen 
parallel  sides,  of  a  regular  hexagon  is  1.732  times  the  side. 

To  convert  Fahrenheit  into  centigrade  subtract  32°  from  the 
Fahrenheit  reading  and  take  ^  of  the  result. 

if 

Expressed  as  a  formula,  C=  g(F—  32). 

To  convert  centigrade  into  Fahrenheit  take  |  of  the  centigrade 
reading  and  add  32°  to  the  result. 
Expressed  as  a  formula,  F=  g  C+  32. 


136  TABLES  AND  RULES 

DECIMAL  EQUIVALENTS  OF  COMMON  FRACTIONS 


Fraction 

Decimal 

Fraction 

Decimal 

1 

6¥ 

0.016 

3  3 
~6¥ 

0.516 

h 

«\ 

.031 
.047 

il 

If 

.531 
.547 

tV 

A 

6\ 

.063 

.078 
.094 

T% 

if 

II 

.563 
.578 
.594 

1 
8 

6^f 

.109 

5 

If 

.609 

.125 

8 

.625 

ii 

.141 

il 

.641 

i^ 

l\ 

.156 
.172 

M 

II 

.656 
.672 

^^ 

^\ 

13 
6  4 

.188 
.203 
.219 

h\ 

II 

II 

.688 
.703 
.719 

1 
4 

64 

.234 

3 
4 

II 

.734 

.250 

.750 

11 

.266 

II 

.766 

ii 

M 

.281 
.297 

If 

n 

.781 
.797 

^ 

M 

li 

.313 
.328 
.344 

if 

II 

ff 

.813 

.828 
.844 

3 
8 

«f 

.359 

7 

If 

.859 

.375 

8 

.875 

If 

.391 

li 

.891 

if 

II 

.406 
.422 

29 
^2 

If 

.906 
.922 

^s 

M 

If 

.438 
.453 
.469 

if 

M 

li 

.938 
.953 
.969 

1 

li 

.484 

If 

.984 

2 

.500 

1 

1.000 

DEFINITIONS 

Terms  Defined.  Students  who  use  a  book  of  this  nature 
will  do  so  m  schools  which  have  shops  in  which  the  machines 
mentioned  in  this  book  either  will  be  found  or  be  described  by 
the  instructor.  Formal  definitions  are  therefore  undesirable. 
In  order  to  save  the  time  of  the  instructor  and  the  student, 
however,  informal  definitions  of  a  few  terms  are  given,  par- 
ticularly such  as  are  not  evident  from  the  blueprints. 

Angle  clamp.  A  clamp  for  holding  work  together  at  an  angle. 

Arbor.  A  shaft  to  hold  Avork  during  some  machine  operation, 
as  on  a  lathe  or  milling  machine. 

Bearing  cap.    The  upper  half  of  a  bearing. 

Bearing  support.  A  casting  used  to  support  a  bearing. 

Binding  post.  A  post  on  an  electric  instrument  to  connect  wires. 

Blind  collar.  A  collar  in  which  the  hole  is  drilled  partly  through. 

Boring  mill.  A  machine  for  boring,  turning,  or  facing. 

Bushing.  A  tube  or  shell  for  reducing  the  diameter  of  a  hole. 

Cartridge  fuse.  A  fuse  used  to  protect  electric  circuits. 

Change  gears.    Gears  used  on  a  lathe  to  drive  the  lead  screw. 

Collar  pin.  A  pin  having  a  collar  and  carrying  a  roll,  gear,  or 
other  part  at  the  outer  end. 

Conduit.    Sherardized  pipe  used  as  protection  to  electric  wires. 

Cone  pulley.  A  pulley  made  up  of  several  pulleys  of  successively 
increasing  sizes  cast  in  one  piece. 

Countershaft.  A  shaft  carrying  tight  and  loose  pulleys  for  start- 
ing and  stopping  machines  or  reversing  their  motion. 

Crankshaft.    The  main  shaft  in  a  gas  engine. 

Diameter.  The  distance  across  a  circle  measured  through  the 
center;  also,  in  workshop  practice,  the  diagonal  of  a  square. 

Drill  socket.  A  device  for  driving  drills  having  a  taper  shank. 

197 


198  DEFINITIONS 

Emery  wheel.  A  grinding  wheel  made  of  emery  or  carborundum. 

Faceplate.   A  disk  on  the  nose  of  a  lathe  for  driving  the  work. 

Feed  screw.  A  threaded  screw  which  gives  movement  to  the 
cross  carriage  of  a  lathe. 

Flanged  coupling.  A  form  of  coupling  used  to  connect  two  shafts. 

Flanged  pulley.  A  pulley  in  which  the  rim  has  a  flange. 

Gear.   A  wheel  with  teeth  on  the  rim  for  transmitting  power. 

Gear  train.  A  series  of  gears  in  which  the  teeth  mesh. 

Gusset  plate.  A  plate  for  strengthening  or  holding  two  or  more 
pieces  at  an  angle. 

Hack  saw.  A  saw  for  cutting  metals. 

I-beam.  A  beam  with  a  cross  section  shaped  like  the  letter  ^^I". 

Jig  bushing.  A  hardened  tool-steel  collar. 

Lathe  spindle.    The  shaft  which  drives  the  work  on  a  lathe. 

Line  shaft.  The  shafting  driving  the  machinery  in  a  shop  by 
means  of  pulleys  and  belting. 

Milling  machine.  A  machine  for  removing  metal  by  means  of 
revolving  cutters. 

Piston.  A  hollow  piece  sliding  in  a  cylinder  and  transmitting 
power  through  a  rod  to  the  crankshaft  of  a  gas  engine. 

Piston  ring.  A  ring  snapped  into  the  grooves  of  a  piston  to 
prevent  the  escape  of  gas. 

Planer.  A  machine  for  producing  plane  surfaces  on  metals. 

Pulley.   A  wheel  for  transmitting  power  by  means  of  a  belt. 

Shaper.  A  machine  for  planing  straight  and  angle  surfaces. 

Spark  plug.  A  device  for  exploding  the  gas  in  the  cylinder  of 
a  gas  engine. 

Spur  gear.  A  toothed  wheel  in  which  the  teeth  are  cut  parallel 
to  the  axis  of  revolution. 

Taper  spindle.  A  tapered  round  shaft. 

Tool  post.   A  post  used  on  a  lathe  to  hold  the  cutting  tool. 

Transmission  case.  The  covering  inclosing  the  gears  for  con- 
trolling the  speed  of  an  automobile. 

Turret  lathe.  A  lathe  with  special  fixtures  for  production  work. 

Web  pulley.  A  pulley  having  a  web,  or  ring  of  metal,  connect- 
ing the  hub  with  the  rim. 


INDEX 


PAGE 

Accuracy  of  measurement    .  6, 

71, 134, 155 

Addition 1,2,14,34 

Amount,  net 47 

Angle 132,194 

of  depression 140 

of  elevation 141 

Apartment-house  structure       .    178 
Applications,  general     .    .     165-192 
miscellaneous  .    13,  33,  38, 46,  58 
Area  .   72,  74,  76,  78,  80,  82,  88, 

96,  97,  98, 116,  119, 126, 195 
Automobile  transmission  .    .    .    176 

Bill 48,50,182,184,186 

Blueprints 

Angle  bracket 89 

clamp 83 

plate 91 

Apartment  house   ....  179 

Automobile  transmission  ,  177 

wheel 95 

Bay  window 153 

Beam 21, 173 

Bearing  cap 101 

support 83 

Binding  post 27 

Blind  collar 101 

Boiler 175 

connections     ....  185 

patch 173 

Bolt,  square-head  ....  5 

Border  of  lamps     ....  35 


PACK 

Blueprints  {continued) 

Boring-mill  table    ....      11 

Bridge 9 

Bushing,  bronze     ....    101 

jig 101 

Cartridge  fuse 7 

Change  gears 67 

Circular  saw 95 

Closet  door 21 

Collar  pin 7 

Compound  rest 89 

Concrete  support    ....    189 
Conduit 25, 183 

wiring 183 

Cone  pulley 3,  191 

Coupling,  flanged   ....    191 

Crankshaft 17 

Cross  slide      189 

Desk 83 

Dining  chair 31 

Door,  closet  ^ 21 

Drill  jig  .    .  ' 153 

socket 29 

Emery  wheel 11, 181 

Feed  screw 17 

Flanged  pulley 191 

shaft-coupling  .  .  .  191 
Floor  plan  ....  69,  173,  179 
Formulas,  useful    ....    119 

Gable  roof 153 

Gear  train 67,  181 

Gusset  plate 83, 173 

Hack  saw 181 


199 


200 


INDEX 


PAGE 

Blueprints  (continuerl) 

I-beam 21 

Intake  pipe 19 

Jig  bushing 101 

Joist 89 

Lamps,  border  of   ...    .  35 

Lathe  faceplate .    .    .    .    7,  191 

gear  box 67 

spindle 3 

Lockers 5, 187 

Milling-machine  arbor  .    .  21 

Pipe  stop 31 

support 189 

Piston 3,  5 

ring 175 

Planer  bolt 21 

gears 67 

table 19 

Pulley  ...    3,  69,  95, 189, 191 

train 181 

Section  of  lockers       .    .    .  187 

Shaft 15, 153 

coupling 191 

support 101 

Shaper  table 189 

Slot  cleaner 89 

Spark  plug 27 

Sprinkler  pipe 37 

Spur-gear  blank     ....  11 

Stairway 9 

Steam  pipe 175 

Steel  bar 35 

girder       37 

Step  block 91 

Stock-room  bins     ....  19 

Stone  pier 91 

Stop  dog 91 

Studs  for  partition     ...  35 

Taper  spindle 29 

Tool  post 15 


PAGE 

Blueprints  (continued) 

Transmission-case  cover    .  37 

Turret-lathe  slide  ....  189 

Useful  formulas     ....  119 

V-thread 153 

Water  pipe 25 

Web  pulley 189 

Wood-shop  floor  plan     .    .  69 

Board  measure 86 

Boiler  connections 184 

Capacity 128, 194 

Carat 42 

Cash  discount 49 

Checks 1,8,136,155 

Circle  .  .  .10,  55,  92,  96,  114, 119 
Circumference  .  .  6,  10,  55,  92, 195 
Circumscribed  circle  ....  120 
Common  equivalents     ....    195 

measures 71,  193 

Complement 138 

Conduit  wiring 182 

Cone 119 

Convenient  rules 195 

Conventional  signs vi 

Cosine 138,  146 

Cotangent 140,  150 

Cube 74,  84, 112 

root 102,112 

Cubic  measure  . 84 

Curve  surface 98 

Cylinder 98 

hollow T   .      99 

Decimals  1,  2,  4,  6,  8, 10,  26,  28, 

40,  104, 157,  196 

Definitions 197 

Diagonal 152,195 

Diameter  .  .  6,  10,  55,  92, 152, 195 
Discount 47,  48,  50 


INDEX 


201 


PAGE 

Discounted  bill 48,  50 

Division     .    .    8,  10,  23,  34, 160, 161 

Drawing  to  scale 56 

Dry  measure 71, 193 

Elevation 88 

Equilateral  triangle  ...     119,  195 

Equivalents 195,196 

Estimating  area     ....       80, 116 

Evaluating 74 

Extremes 59 

Feet 34 

per  minute  (F.  P.  M.)  .    .  93, 188 
Formulas    74,  76,  78,  84,  86,  92, 
96,97,98,99,105,114,118, 

119,  132,  135, 138, 140, 195 

Fourth  root 102 

Fractions    14, 16, 18,  20,  22,  23, 

24,  26,  28,  40,  47,  71, 196 

Function 142 

Fundamental  operations   ...        1 

Gear 63 

train 64, 180 

General  applications     .    .     165-192 
Gram 123,128 

Hexagon 119,195 

Hollow  cylinder 99 

Horse  power 41 

Hypotenuse 105 

Idler 66,  68 

Inches 34 

Index  of  a  root 102 

Indirect  measurement  ....    131 

Infinity 133 

Inscribed  circle 120 

Intermediate  gear     .    .    .    .    66, 68 


PAGE 

Interpolation 142 

Inverse  proportion 62 

Invoice 48 

Length 71, 124, 193, 194 

Liquid  measure 71, 193 

List  price 47 

Liter •    .    •     123,128 

Load 87 

Locker  section 186 

Lumber 86 

Maps 56 

Means 69 

Measurement,  accuracy  of    .  6, 

71, 134,  155 

Mensuration 71 

Meter 13,123,124 

Metric  system 122, 194 

Miscellaneous  applications    13, 

33,  38,  46,  58 
Multiplication     4,  6, 18,  20,  22, 

157,  159, 161 

Natural  functions 142 

Net  amount 47 

price 47 

Octagon 77,  119 

Overhead  charges 54 

Pantograph 60 

Parallelogram 76 

Pay  roll 62 

Per  cent 40 

Percentage 40,  42,  44 

Perch 90 

Perimeter 13 

TT  (pi,  3|,  3.1416)  6, 10, 13,  55,  92, 158 
Pitch 4, 162, 165 


202 


INDEX 


PAGE 

Price,  list 47 

net 47 

selling 47 

Prime  cost 54 

Prism 86, 119 

Problem  material  {see  Blue- 
prints, General  applications. 
Miscellaneous  applications. 
Reviews,  and  subject  titles) 

Proportion 59, 162 

Pulley 62 

train 64, 180 

Radius 10, 114,  163 

Rate  of  discount 47 

Ratio 55, 132 

Rectangle 72,  74, 119 

Rectangular  solid 84 

Reduction  of  fractions      .  14,  18,  26 
Reviews      12,  30,  32,  36,  54,  70, 
115, 121, 130,  152, 

164, 165, 188 
Revolutions  per  minute  (R.P.M.)  62 
Right  triangle    .    .     .     105,  119,  132 

Ring,  circular 97,  ll9 

Root 102 

Rules,  convenient 195 

Scale 56 

Selling  price 47 

Several  discounts 50 

Significant  figure 102 

Sine 135,144 

Slide  rule 155 

Specific  gravity      .....  58,  128 

Sphere 119 

Square   ...     74,  112,  119,  163,  195 
measure      ...     .72, 126,  193 

root     .......     102, 163 

Squared  paper  .    .    80, 116, 133, 135 


PAGE 

Stock  bill 182, 184, 186 

Subtraction 1,  2, 16 

Symbols  vi,  2,  28,  40,  51,  53,  65, 

71,  90, 102,  133, 137,  142 

Tables    .     108-112,  142-151, 193-196 

angles  and  arcs 194 

common  measures      .    .    .    193 

cosines 138, 146 

cotangents 140, 150 

counting 194 

cube  roots 112 

cubes 112 

decimals 196 

fractions 196 

functions 142 

metric     .    .       124,126,128,194 

per  cents 40 

powers  and  roots    ....    112 

sines 135,136,144 

square  roots   ....      107-112 
squares  ........    112 

tangents 133,  148 

Tangent 132,  148 

Terms  of  discount 48 

of  a  ratio 55 

Trade  discount 48 

Transmission,  automobile     .    .    176 

Trapezoid 78 

Triangle     .    .     76,105,119,131,195 
equilateral      ....     119, 195 

right 105, 119,  132 

Trigonometry 131 

Volume     84,  86,  88,  90,  98,  119, 

126,  193 

Weight      .    .      71,128,193,194,195 
Working  drawings   (see   Blue- 
prints)      82 


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